Cup products, lower central series, and holonomy Lie algebras

TL;DR

This paper extends the relationship between graded Lie algebras and holonomy Lie algebras to all finitely presented groups, providing explicit formulas and algorithms for computation, with applications in topology and group theory.

Contribution

It generalizes key results to arbitrary finitely presented groups and introduces an explicit formula and an algorithm for computing holonomy Lie algebras.

Findings

Explicit formula for cup-product in cohomology of finite 2-complex

Algorithm for computing holonomy Lie algebra using Magnus expansion

Applications to link groups, one-relator groups, and Seifert fibered manifolds

Abstract

We generalize basic results relating the associated graded Lie algebra and the holonomy Lie algebra from finitely presented, commutator-relators groups to arbitrary finitely presented groups. In the process, we give an explicit formula for the cup-product in the cohomology of a finite 2-complex, and an algorithm for computing the corresponding holonomy Lie algebra, using a Magnus expansion method. We illustrate our approach with examples drawn from a variety of group-theoretic and topological contexts, such as link groups, one-relator groups, and fundamental groups of orientable Seifert fibered manifolds.

Equations

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$$\mathbb{Q}F\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$u_i\cup u_j=\sum _{k=n-b+1}^m\kappa (w_k{)}_{i,j}{\beta }_k,\text{for 1≤i,j≤b}.$$

$$\mathrm{gr}(G):=\underset{k\ge 1}{⨁}\left({\Gamma }_{k}G/{\Gamma }_{k+1}G\right)\otimes \mathbb{Q},$$

$$M(x_i)=1+x_i\text{and}M(x_i^{-1})=1-x_i+x_i^2-x_i^3+\cdots .$$

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$$M_2(y)=\sum _{i<j}{\epsilon }_{i,j}(y)(x_i{x}_j-x_j{x}_i).$$

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$$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 10.79863pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&\crcr}}}\ignorespaces{\hbox{\kern-10.79863pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\mathbb{Q}{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 23.1715pt\raise 5.39166pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.39166pt\hbox{$\scriptstyle{M}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 50.79863pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 50.79863pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\widehat{T}(F_{\mathbb{Q}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 91.82268pt\raise 7.48611pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.73611pt\hbox{$\scriptstyle{\widehat{T}({\pi})}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 122.30702pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 122.30702pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\widehat{T}(G_{\mathbb{Q}})}$}}}}}}}\ignorespaces}}}}\ignorespaces,$$

$$\kappa (r)=1+\sum _I\kappa (r{)}_I{y}_I.$$

$$\mathbb{Q}F\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\kappa (uv{)}_I=\kappa (u{)}_I+\kappa (v{)}_I,\text{for ∣I∣=s}.$$

$$\kappa (uv)=\kappa (u)\kappa (v)=1+\sum _{|I|=s}(\kappa (u{)}_I+\kappa (v{)}_I)y_I+\text{higher-order terms}.$$

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$${\kappa }_2(r)=\sum _{1\le i<j\le b}\kappa (r{)}_{i,j}(y_i{y}_j-y_j{y}_i).$$

$$\kappa (r{)}_i=\sum _{s=1}^n{a}_{i,s}{\epsilon }_s(r)\text{ and }\kappa (r{)}_{i,j}=\sum _{s,t=1}^n{a}_{i,s}{a}_{j,t}{\epsilon }_{s,t}(r).$$

$${\kappa }_1(r)=\pi \circ M_1(r)=\pi \left(\sum _{s=1}^n{\epsilon }_s(r)x_s\right)=\sum _{s=1}^n\sum _{i=1}^b{a}_{i,s}{\epsilon }_s(r)y_i,$$

$${\kappa }_2(r)=\pi \otimes \pi \circ M_2(r)=\pi \otimes \pi \left(\sum _{s,t=1}^n{\epsilon }_{s,t}(r)x_s\otimes x_t\right)=\sum _{s,t=1}^n\sum _{i,j=1}^b{\epsilon }_{s,t}(r)a_{i,s}{a}_{j,t}{y}_i\otimes y_j,$$

$${\tilde{d}}_1({\tilde{e}}_i^1)=x_i-1,{\tilde{d}}_2({\tilde{e}}_j^2)=\sum _{k=1}^m\phi ({\partial }_k{r}_j){\tilde{e}}_k^1,$$

$$G_e=⟨x_1,\dots ,x_n\mid w_1,\dots ,w_m⟩,$$

$${\gamma }_k:=f_{\ast }(e_k^2)=\sum _{l=1}^m{c}_{l,k}{z}_l,$$

$${\delta }_p[g_1|\dots |g_p]=g_1[g_2|\dots |g_p]+\sum _{i=1}^{p-1}(-1{)}^i[g_1|\dots |g_i{g}_{i+1}|\dots |g_p]+(-1{)}^p[g_1|\dots |g_{p-1}].$$

$$u\cup u^{\prime }[g_1|g_2]=u(g_1)u^{\prime }(g_2).$$

$$0$$

$$T_1({\tilde{e}}_i^1)=[x_i]and{T}_2({\tilde{e}}_k^2)={\tau }_1{T}_1{\tilde{d}}_2({\tilde{e}}_k^2),$$

$${\tau }_1(g[g_1])=[g|g_1],$$

$$u_i([\phi (r)])=\sum _{s=1}^n{\epsilon }_s(r)a_{i,s}={\kappa }_i(r),$$

$$u_i([\phi (r)])=\sum _{s=1}^n{\epsilon }_s(r)u_i([x_s])=\sum _{s=1}^n{\epsilon }_s(r)a_{i,s}={\kappa }_i(r).$$

$${\partial }_t(w_k)=\sum _{x\in F}{p}_{tk}^{x}x,$$

Full text

1Supported in part by the National Security Agency (grant H98230-13-1-0225) and the Simons Foundation (collaboration grant for mathematicians 354156)

1. 1 Introduction
2. 2 Expansions for finitely presented groups
3. 3 Echelon presentations and cellular chain complexes
4. 4 Group presentations and (co)homology
5. 5 A presentation for the holonomy Lie algebra
6. 6 Lower central series and the holonomy Lie algebra
7. 7 Mildness and graded-formality
8. 8 One-relator groups
9. 9 Seifert fibered spaces

1. Introduction

Throughout this paper $G$ will be a finitely generated group. Our main focus will be on the cup-product in the rational cohomology of the $2$-complex associated to a presentation of $G$, and on several rational Lie algebras attached to such a group.

1.1. Magnus expansions and cup products

The notion of expansion of a group, which goes back to W. Magnus [21], has been generalized and used in many ways. For instance, a presentation for the Malcev Lie algebra of a finitely presented group was given by S. Papadima [30] and G. Massuyeau [25], while X.-S. Lin [20] studied expansions of fundamental groups of smooth manifolds. Recently, D. Bar-Natan [2] has generalized the notion of expansion and has introduced the Taylor expansion of an arbitrary ring. In turn, we explored in [41] various relationships between expansions and formality properties of groups.

We go back here to the classical Magnus expansion, and adapt it for our purposes. Let $G$ be a group with a finite presentation $G=F/R=\langle x_{1},\dots,x_{n}\mid r_{1},\dots,r_{m}\rangle$. There exists then a $2$-complex $K=K_{G}$ associated to such a presentation. In the case when $G$ is a commutator-relators group, i.e., when all relators $r_{i}$ belong to the commutator subgroup $[F,F]$, R. Fenn and D. Sjerve computed in [10] the cup-product map

[TABLE]

using the Magnus expansion $M\colon\mathbb{Z}{F}\to\mathbb{Z}\langle\!\langle\mathbf{x}\rangle\!\rangle$ from the group ring of the free group $F=\langle x_{1},\dots,x_{n}\rangle$ to the power series ring in $n$ non-commuting variables, which is the ring morphism defined by $M(x_{i})=1+x_{i}$.

Our first objective in this work is to generalize this result of Fenn and Sjerve, from commutator-relators groups to arbitrary finitely presented groups. We will avoid possible torsion in the first homology of $G$ by working over the field of rationals. To that end, we start by defining a Magnus-like expansion $\kappa=\kappa_{G}$ relative to such a group $G$ as the composition

[TABLE]

where $\varphi\colon F\twoheadrightarrow G$ is the canonical projection and $\widehat{T}(V)$ is the completed tensor algebra of a vector space $V$. We then show in Proposition 3.3 that there exists a group $G_{e}$ admitting a ‘row-echelon’ presentation, $G_{e}=\langle x_{1},\dots,x_{n}\mid w_{1},\dots,w_{m}\rangle$, and a map $f\colon K_{G_{e}}\to K_{G}$ inducing an isomorphism in cohomology.

Using the $\kappa$-expansion of $G_{e}$, we determine in Theorem 4.3 the cup-product map for $K_{G_{e}}$, from which we obtain in Theorem 4.4 a formula for computing the cup-product map $\mu_{K}$, with $\mathbb{Q}$-coefficients. Let $b=b_{1}(G)$ be the first Betti number of $G$, and let $\{u_{1},\dots,u_{b}\}$ and $\{\beta_{n-b+1},\dots,\beta_{m}\}$ be bases for $H^{1}(K;\mathbb{Q})$ and $H^{2}(K;\mathbb{Q})$, transferred from suitable bases in the rational cohomology of $K_{G_{e}}$ via the isomorphism $f^{*}\colon H^{*}(K_{G_{e}};\mathbb{Q})\to H^{*}(K_{G};\mathbb{Q})$. Our result then reads as follows.

Theorem 1.1**.**

Let $K$ be a presentation $2$-complex for a finitely presented group $G$. In the bases described above, the cup-product map $\mu_{K}\colon H^{1}(K;\mathbb{Q})\wedge H^{1}(K;\mathbb{Q})\rightarrow H^{2}(K;\mathbb{Q})$ is given by

[TABLE]

Let us note that the map $\mu_{K}$ depends on the chosen presentation for $G$, and may differ from the cup-product map $\mu_{G}$ in a classifying space for $G$. However, the two maps share the same kernel, which makes the algorithm for determining the map $\mu_{K}$ useful in other contexts, for instance, in computing the first resonance variety of $G$ (see e.g. [27, 35]), or finding a presentation for the holonomy Lie algebra $\mathfrak{h}(G)$, a procedure that we discuss next.

1.2. Holonomy Lie algebras

The holonomy Lie algebra of a finitely generated group $G$, denoted by $\mathfrak{h}(G)$, is the quotient of the free Lie algebra on $H_{1}(G;\mathbb{Q})$ by the Lie ideal generated by the image of the dual of the cup-product map $\mu_{G}$. It is easy to see that $\mathfrak{h}(G)$ is a graded Lie algebra over $\mathbb{Q}$ which admits a quadratic presentation depending only on $\ker\mu_{G}$. Moreover, this construction is functorial. The holonomy Lie algebra was introduced by T. Kohno in [13], building on work of Chen [6], and has been further studied in a number of papers, including [24, 31, 41].

Our next objective is to find a presentation for the holonomy Lie algebra $\mathfrak{h}(G)$. We start by showing in Proposition 5.4 that there is a homomorphism from a finitely presented group $G_{f}$ to $G$ inducing an isomorphism on holonomy Lie algebras. Hence, without loss of generality, we may assume that $G$ admits a finite presentation.

Let $G_{e}=\langle x_{1},\dots,x_{n}\mid w_{1},\dots,w_{m}\rangle$ be a group with row-echelon presentation, and let $\rho\colon G_{e}\to G$ be the homomorphism induced on fundamental groups by the aforementioned map, $f\colon K_{G_{e}}\to K_{G}$. It follows from Corollary 5.3 that the induced map, $\mathfrak{h}(\rho)\colon\mathfrak{h}(G_{e})\to\mathfrak{h}(G)$, is an isomorphism of graded Lie algebras. Using the computation of the cup-product map $\mu_{K_{G}}$ from Theorem 1.1, we describe in Theorem 5.5 an algorithm for finding a presentation for the holonomy Lie algebra $\mathfrak{h}(G)$. Furthermore, we obtain in Theorem 5.11 a presentation for the derived quotients of this Lie algebra, $\mathfrak{h}(G)/\mathfrak{h}(G)^{(i)}$. Our results may be summarized as follows.

Theorem 1.2**.**

Let $G$ be a finitely presented group. The holonomy Lie algebra $\mathfrak{h}(G)$ is the quotient of the free $\mathbb{Q}$-Lie algebra with generators $\mathbf{y}=\{y_{1},\dots,y_{b}\}$ in degree $1$ by the ideal $I$ generated by $\kappa_{2}(w_{n-b+1}),\dots,\kappa_{2}(w_{m})$, where $\kappa_{2}$ is the degree $2$ part of the Magnus expansion of $G_{e}$. Furthermore, for each $i\geq 2$, the solvable quotient $\mathfrak{h}(G)/\mathfrak{h}(G)^{(i)}$ is isomorphic to $\mathfrak{lie}(\mathbf{y})/(I+\mathfrak{lie}^{(i)}(\mathbf{y}))$.

In the special case when $G$ admits a presentation with only commutator relators, presentations for these Lie algebras were given by Papadima and Suciu in [31]. For arbitrary finitely generated groups $G$, the metabelian quotient $\mathfrak{h}(G)/\mathfrak{h}(G)^{\prime\prime}$, also known as the holonomy Chen Lie algebra of $G$, is closely related to the first resonance variety of $G$, a geometric object which has been studied intensely from many points of view, see for instance [27, 34, 35, 43, 44] and references therein.

1.3. Lower central series, graded formality, and mildness

The Lie methods in group theory were introduced by W. Magnus in [22], and further developed by E. Witt [48], M. Hall [12], M. Lazard [18], and many more authors, see for instance [23]. The associated graded Lie ring of a group $G$ is the graded Lie ring $\operatorname{gr}(G;\mathbb{Z})$, whose graded pieces are the successive quotients of the lower central series of $G$, and whose Lie bracket is induced from the group commutator. The quintessential example is the associated graded Lie ring of the free group on $n$ generators, $F_{n}$, which is isomorphic to the free Lie ring $\mathfrak{lie}(\mathbb{Z}^{n})$. Much of the power of this method comes from the various connections between lower central series, nilpotent quotients, and group homology, as evidenced in the work of J. Stallings [40], W. Dwyer [8], and many others.

We concentrate here on the associated graded Lie algebra over the rationals, $\operatorname{gr}(G)=\operatorname{gr}(G;\mathbb{Z})\otimes\mathbb{Q}$, of a finitely generated group $G$. As we recall in §6.2, there is a natural epimorphism $\Phi_{G}\colon\mathfrak{h}(G)\twoheadrightarrow\operatorname{gr}(G)$. Thus, the holonomy Lie algebra $\mathfrak{h}(G)$ may be viewed as a quadratic approximation to the associated graded Lie algebra of $G$. We say that the group $G$ is graded-formal if the map $\Phi_{G}$ is an isomorphism of graded Lie algebras. A much stronger requirement is that $G$ be $1$-formal, a condition we recall in §2.1. For much more on these notions, we refer to [31, 32, 41].

In Propositions 6.2 and 6.3, we compare the holonomy Lie algebra of $G$ with the holonomy Lie algebras of the nilpotent quotients $G/\Gamma_{i}G$ and the derived quotients $G/G^{(i)}$. In Corollary 6.6, we use Theorem 1.2 and a result from [41] to give explicit presentations for the graded Lie algebras \operatorname{gr}\big{(}G/G^{(i)}\big{)} in the case when $G$ is a finitely presented, $1$-formal group.

Some of the motivation for our study comes from the work of J. Labute [14, 15] and D. Anick [1], who gave presentations for the associated graded Lie algebra $\operatorname{gr}(G)$ in the case when $G$ is ‘mildly’ presented. We revisit this topic in §7, where we relate the notion of mild presentation to that of graded formality, and derive some consequences, especially in the context of link groups.

1.4. Further applications

We illustrate our approach with several classes of finitely presented groups, including $1$-relator groups in §8, and fundamental groups of orientable Seifert fibered $3$-manifolds with orientable base in §9. We give here presentations for the holonomy Lie algebra $\mathfrak{h}(G)$ and the Chen Lie algebra $\operatorname{gr}(G/G^{\prime\prime})$ of such groups $G$. We also compute the Hilbert series of these graded Lie algebras, and discuss the formality properties of these groups.

This work was motivated by a desire to generalize some of the results of Fenn–Sjerve [10] and Papadima–Suciu [31], from commutator-relators groups to arbitrary finitely generated groups. In [41], we studied the formality properties of finitely generated groups, focusing on the filtered-formality and $1$-formality properties. In related work, we apply the techniques developed in this paper and in [41] to the study of several families of “pure-braid like” groups. For instance, we investigate in [43] the pure virtual braid groups, and we investigate in [44] the McCool groups, also known as the pure welded braid groups. A summary of these results, as well as further motivation and background can be found in [42].

2. Expansions for finitely presented groups

In this section, we introduce and study a Magnus-type expansion relative to a finitely presented group. We start by reviewing some basic notions.

2.1. Completed group algebras and expansions

Let $G$ be a finitely generated group. As is well-known (see for instance [36, 37]), the group-algebra $\mathbb{Q}{G}$ has a natural Hopf algebra structure, with comultiplication $\Delta\colon\mathbb{Q}{G}\to\mathbb{Q}{G}\otimes_{\mathbb{Q}}\mathbb{Q}{G}$ given by $\Delta(g)=g\otimes g$ for $g\in G$, and counit the augmentation map $\varepsilon\colon\mathbb{Q}G\rightarrow\mathbb{Q}$ given by $\varepsilon(g)=1$. The powers of the augmentation ideal, $I=\ker\varepsilon$, form a descending, multiplicative filtration of $\mathbb{Q}{G}$. The associated graded algebra, ${\operatorname{gr}}(\mathbb{Q}{G})=\bigoplus_{k\geq 0}I^{k}/I^{k+1}$, comes endowed with the degree filtration, $\mathscr{F}_{k}=\bigoplus_{j\geq k}I^{j}/I^{j+1}$. The corresponding completion, $\widehat{\operatorname{gr}}(\mathbb{Q}{G})$, is again an algebra, endowed with the inverse limit filtration.

The $I$-adic completion of the group-algebra, $\widehat{\mathbb{Q}{G}}=\varprojlim_{k}\mathbb{Q}{G}/I^{k}$, also comes equipped with an inverse limit filtration. Applying the $I$-adic completion functor to the map $\Delta$ yields a comultiplication map $\widehat{\Delta}$, which makes $\widehat{\mathbb{Q}{G}}$ into a complete Hopf algebra, see [38, App. A].

An element $x$ in a Hopf algebra is called primitive if $\Delta x=x{\otimes}1+1{\otimes}x$. The set $\mathfrak{m}(G)$ of all primitive elements in $\widehat{\mathbb{Q}{G}}$, with bracket $[x,y]=xy-yx$, is a complete, filtered Lie algebra, called the Malcev Lie algebra of $G$. The set of all primitive elements in $\operatorname{gr}(\mathbb{Q}G)$ forms a graded Lie algebra, which is isomorphic to the associated graded Lie algebra

[TABLE]

where $\{\Gamma_{k}G\}_{k\geq 1}$ is the lower central series of $G$, defined inductively by $\Gamma_{1}G=G$ and $\Gamma_{k+1}G=[\Gamma_{k}G,G]$ for $k\geq 1$. As shown by Quillen in [37], there is an isomorphism of graded Lie algebras, $\operatorname{gr}(\mathfrak{m}(G))\cong\operatorname{gr}(G)$.

The group $G$ is said to be filtered-formal if its Malcev Lie algebra is isomorphic (as a filtered Lie algebra) to the degree completion of its associated graded Lie algebra. The group $G$ is said to be $1$-formal if its Malcev Lie algebra admits a quadratic presentation (see [32, 41] for more details). For instance, all finitely generated free groups and free abelian groups are $1$-formal.

An expansion for a group $G$ is a filtration-preserving algebra morphism ${E}\colon\mathbb{Q}{G}\to\widehat{\operatorname{gr}}(\mathbb{Q}{G})$ with the property that $\operatorname{gr}({E})=\operatorname{id}$ (see [20, 2, 41]). As shown in [41], a finitely generated group $G$ is filtered-formal if and only if it has an expansion $E$ which induces an isomorphism of complete Hopf algebras, $\widehat{E}\colon\widehat{\mathbb{Q}{G}}\to\widehat{\operatorname{gr}}(\mathbb{Q}{G})$.

2.2. The Magnus expansion for a free group

Let $F$ be a finitely generated free group, with generating set $\mathbf{x}=\{x_{1},\dots,x_{n}\}$, and let $\mathbb{Z}{F}$ be its group-ring. Then the degree completion of the associated graded ring, $\widehat{\operatorname{gr}}(\mathbb{Z}{F})$, can be identified with the completed tensor ring $\widehat{T}(F_{\operatorname{ab}})=\mathbb{Z}\langle\!\langle\mathbf{x}\rangle\!\rangle$, the power series ring over $\mathbb{Z}$ in $n$ non-commuting variables, by sending $[x_{i}-1]$ to $x_{i}$. There is a well known expansion $M\colon\mathbb{Z}{F}\to\mathbb{Z}\langle\!\langle\mathbf{x}\rangle\!\rangle$, called the Magnus expansion, given by

[TABLE]

The Fox derivatives are the ring morphisms $\partial_{i}\colon\mathbb{Z}{F}\rightarrow\mathbb{Z}{F}$ defined by the rules $\partial_{i}(1)=0$, $\partial_{i}(x_{j})=\delta_{ij}$, and $\partial_{i}(uv)=\partial_{i}(u)\varepsilon(v)+u\partial_{i}(v)$ for $u,v\in\mathbb{Z}{F}$, where $\varepsilon\colon\mathbb{Z}{F}\to\mathbb{Z}$ is the augmentation map. The higher Fox derivatives $\partial_{i_{1},\dots,i_{k}}$ are then defined inductively. We refer to [9, 10, 23, 27] for more details and references on these notions.

The Magnus expansion can be computed in terms of the Fox derivatives, as follows. Given an element $y\in F$, if we write $M(y)=1+\sum a_{I}x_{I}$, then $a_{I}=\varepsilon_{I}(y)$, where $I=(i_{1},\dots,i_{s})$, and $\varepsilon_{I}=\varepsilon\circ\partial_{I}$ is the composition of the augmentation map with the iterated Fox derivative $\partial_{I}\colon\mathbb{Z}{F}\to\mathbb{Z}{F}$. For each $k\geq 1$, let $M_{k}$ be the composite

[TABLE]

For each $y$ in $F$, we have that $M_{1}(y)=\sum_{i=1}^{n}\varepsilon_{i}(y)x_{i}$, while for each $y$ in the commutator subgroup $[F,F]$, we have

[TABLE]

The tensor algebra $T(F_{\mathbb{Q}})$ on the $\mathbb{Q}$-vector space $F_{\mathbb{Q}}=F_{\operatorname{ab}}\otimes\mathbb{Q}$ has a natural graded Hopf algebra structure, with comultiplication $\Delta$ and counit $\varepsilon$ given by $\Delta(a)=a\otimes 1+1\otimes a$ and $\varepsilon(a)=0$ for $a\in F_{\mathbb{Q}}$. The set of primitive elements in $T(F_{\mathbb{Q}})$ is the free Lie algebra $\mathfrak{lie}(F_{\mathbb{Q}})=\{v\in T(F_{\mathbb{Q}})\mid\Delta(v)=v\otimes 1+1\otimes v\}$, with Lie bracket $[v,w]=v\otimes w-w\otimes v$. Notice that, if $y\in[F,F]$, then $M_{2}(y)$ is a primitive element in the degree $2$ piece of the Hopf algebra ${T}(F_{\mathbb{Q}})=\operatorname{gr}(\widehat{T}(F_{\mathbb{Q}}))$, which corresponds to the degree $2$ element $\sum_{i<j}\varepsilon_{i,j}(y)[x_{i},x_{j}]$ in the free Lie algebra $\mathfrak{lie}(F_{\mathbb{Q}})$.

2.3. The Magnus expansion relative to a finitely generated group

Given a finitely generated group $G$, there exists an epimorphism $\varphi\colon F\twoheadrightarrow G$ from a free group $F$ of finite rank to $G$. Let $\varphi_{\operatorname{ab}}\colon F_{\operatorname{ab}}\twoheadrightarrow G_{\operatorname{ab}}$ be the induced epimorphism between the respective abelianizations.

Definition 2.1**.**

The Magnus $\kappa$-expansion for $F$ relative to $G$, denoted by $\kappa_{G}$ (or $\kappa$ for short), is the composition

[TABLE]

where $M$ is the classical Magnus expansion for the free group $F$, and the morphism $\widehat{T}(\varphi_{\operatorname{ab}})\colon\widehat{T}(F_{\operatorname{ab}})\twoheadrightarrow\widehat{T}(G_{\operatorname{ab}})$ is induced by the abelianization map $\varphi_{\operatorname{ab}}$.

In particular, if the group $G$ is a commutator-relators group, i.e., if all the relators of $G$ lie in the commutator subgroup $[F,F]$, then the projection $\varphi_{\operatorname{ab}}$ identifies $F_{\operatorname{ab}}$ with $G_{\operatorname{ab}}$, and the Magnus expansion $\kappa$ coincides with the classical Magnus expansion $M$.

More generally, let $G$ be a group generated by $\mathbf{x}=\{x_{1},\dots,x_{n}\}$, and let $F$ be the free group generated by the same set. The rational Magnus $\kappa$-expansion, still denoted by $\kappa_{G}$ or $\kappa$, is the composition

[TABLE]

where $\pi=\varphi_{\operatorname{ab}}\otimes\mathbb{Q}=H_{1}(\varphi,\mathbb{Q})$ is the induced epimorphism in homology from $F_{\mathbb{Q}}:=H_{1}(F;\mathbb{Q})$ to $G_{\mathbb{Q}}:=H_{1}(G;\mathbb{Q})$. Pick a basis $\mathbf{y}=\{y_{1},\dots,y_{b}\}$ for $G_{\mathbb{Q}}$, and identify $\widehat{T}(G_{\mathbb{Q}})$ with $\mathbb{Q}\langle\!\langle\mathbf{y}\rangle\!\rangle$. Let $\kappa(r)_{I}$ be the coefficient of $y_{I}:=y_{i_{1}}\cdots y_{i_{s}}$ in $\kappa(r)$, for $I=(i_{1},\dots,i_{s})$. Then we can write

[TABLE]

Lemma 2.2**.**

If $r\in\Gamma_{k}F$, then $\kappa(r)_{I}=0$, for $\left|I\right|<k$. Furthermore, if $r\in\Gamma_{2}F$, then $\kappa(r)_{i,j}=-\kappa(r)_{j,i}$.

Proof.

Since $M(r)_{I}=\varepsilon_{I}(r)=0$ for $\left|I\right|<k$ (see for instance [27]), we have that $\kappa(r)_{I}=0$ for $\left|I\right|<k$. To prove the second assertion, identify the completed symmetric algebras $\widehat{\operatorname{Sym}}(F_{\mathbb{Q}})$ and $\widehat{\operatorname{Sym}}(G_{\mathbb{Q}})$ with the power series rings $\mathbb{Q}[\![\mathbf{x}]\!]$ and $\mathbb{Q}[\![\mathbf{y}]\!]$, respectively, in the following commuting diagram of $\mathbb{Q}$-linear maps.

[TABLE]

When $r\in[F,F]$, we have that $\alpha_{2}\circ\kappa(r)=\widehat{\operatorname{Sym}}(\pi)\circ\alpha_{1}\circ M(r)=1$. Thus, $\kappa_{i}(r)=0$ and $\kappa(r)_{i,j}+\kappa(r)_{j,i}=0$. ∎

Lemma 2.3**.**

If $u,v\in F$ satisfy $\kappa(u)_{J}=\kappa(v)_{J}=0$ for all $\left|J\right|<s$, for some $s\geq 2$, then

[TABLE]

Moreover, the above formula is always true for $s=1$.

Proof.

We have that $\kappa(uv)=\kappa(u)\kappa(v)$ for $u,v\in F$. If $\kappa(u)_{J}=\kappa(v)_{J}=0$ for all $|J|<s$, then $\kappa(u)=1+\sum_{\left|I\right|=s}\kappa(u)_{I}y_{I}$ up to higher-order terms, and similarly for $\kappa(v)$. Then

[TABLE]

Therefore, $\kappa(uv)_{i}=\kappa(u)_{i}+\kappa(v)_{i}$, and so $\kappa(uv)_{I}=\kappa(u)_{I}+\kappa(v)_{I}$. ∎

2.4. Truncating the Magnus expansions

Recall that we defined in (5) truncations $M_{k}$ of the Magnus expansion $M$ of a free group $F$. In a similar manner, we can also define the truncations of the Magnus expansion $\kappa$ for any finitely generated group $G$.

Lemma 2.4**.**

For each $k\geq 1$, the following diagram commutes:

[TABLE]

Proof.

The triangle on the left of diagram (12) commutes, since it consists of ring morphisms, by the definition of the Magnus expansion for a group. The morphisms in the two squares are homomorphisms between $\mathbb{Q}$-vector spaces. The squares commute, since $\pi$ is a linear map. ∎

In diagram (12), let us denote the composition of $\kappa$ and $\operatorname{gr}_{k}$ by $\kappa_{k}$. We then obtain the diagram

[TABLE]

In particular, $\kappa_{1}(r)=\sum_{i=1}^{b}\kappa(r)_{i}y_{i}$ for $r\in F$. By Lemma 2.2, if $r\in[F,F]$, then

[TABLE]

Notice that $\kappa_{2}(r)$ is a primitive element in the Hopf algebra ${T}(G_{\mathbb{Q}})$, which corresponds to the element $\sum_{i<j}\kappa_{i,j}(r)[y_{i},y_{j}]$ in the free Lie algebra $\mathfrak{lie}(G_{\mathbb{Q}})$.

The next lemma provides a close connection between the Magnus expansion $\kappa$ and the classical Magnus expansion $M$.

Lemma 2.5**.**

Let $(a_{i,s})$ be the $b\times n$ matrix associated to the linear map $\pi\colon F_{\mathbb{Q}}\rightarrow G_{\mathbb{Q}}$, and let $r\in F$ be an arbitrary element. Then, for each $1\leq i,j\leq b$, we have that

[TABLE]

Proof.

By assumption, $\pi(x_{s})=\sum_{i=1}^{b}a_{i,s}y_{i}$. By Lemma 2.4 (for $k=1$), we have

[TABLE]

which gives the claimed formula for $\kappa(r)_{i}$. By Lemma 2.4 (for $k=2$), we have

[TABLE]

which gives the claimed formula for $\kappa(r)_{i,j}$. ∎

3. Echelon presentations and cellular chain complexes

In this section we associate to every finitely presented group $G$ an “echelon approximation”, $G_{e}$, such that they have isomorphic cohomology on their respective $2$-complexes.

3.1. Presentation $2$-complex

We start with a brief review of the cellular chain complexes associated to a presentation $2$-complex of a group, following the exposition from [4, 9, 10, 27, 33]. Let $G$ be a group with a finite presentation $P=\langle\mathbf{x}\mid\mathbf{r}\rangle$, where $\mathbf{x}=\{x_{1},\dots,x_{n}\}$ and $\mathbf{r}=\{r_{1},\dots,r_{m}\}$. Then $G=F/R$, where $F$ is the free group on generating set $\mathbf{x}$ and $R$ is the (free) subgroup of $F$ normally generated by the set $\mathbf{r}\subset F$.

Let $K_{P}$ be the $2$-complex associated to this presentation of $G$, consisting of a [math]-cell $e^{0}$, one-cells $\{e_{1}^{1},\cdots,e_{n}^{1}\}$ corresponding to the generators, and two-cells $\{e_{1}^{2},\dots,e_{m}^{2}\}$ corresponding to the relators. The $2$-complex $K_{P}$ depends on the presentation $P$ for the group $G$. However, if the presentation is understood, we may also denote this $2$-complex by $K_{G}$.

The (integral) cellular chain complex $C_{*}=C_{*}(K_{P};\mathbb{Z})$ is of the form $C_{2}\xrightarrow{d_{2}}C_{1}\xrightarrow{d_{1}}C_{0}$, where $C_{j}$ are the free abelian groups on the specified bases. Furthermore, $d_{1}=0$, while the matrix of the boundary map $d_{2}\colon C_{2}(K_{P};\mathbb{Z})\rightarrow C_{1}(K_{P};\mathbb{Z})$ is the $m\times n$ Jacobian matrix $J_{P}=(\varepsilon_{i}(r_{k}))$.

Next, let $p\colon\widetilde{K_{P}}\to K_{P}$ be the universal cover of the presentation $2$-complex, and fix a lift $\tilde{e}^{0}$ of the basepoint $e^{0}$. The cells $e^{i}_{j}$ of $K_{P}$ lift to cells $\tilde{e}^{i}_{j}$ at the basepoint $\tilde{e}^{0}$. Let $\widetilde{C}_{*}=C_{*}(\widetilde{K_{P}};\mathbb{Z})$ be the (equivariant) cellular chain complex of the universal cover. This is a chain complex of free $\mathbb{Z}{G}$-modules of the form $\widetilde{C}_{2}\xrightarrow{\tilde{d}_{2}}\widetilde{C}_{1}\xrightarrow{\tilde{d}_{1}}\widetilde{C}_{0}$, with $\widetilde{C}_{0}=\mathbb{Z}{G}$, $\widetilde{C}_{1}=(\mathbb{Z}{G})^{n}$ generated by the set $\{\tilde{e}^{1}_{1},\dots,\tilde{e}^{1}_{n}\}$, and $\widetilde{C}_{2}=(\mathbb{Z}{G})^{m}$ generated by the set $\{\tilde{e}^{2}_{1},\dots,\tilde{e}^{2}_{m}\}$. The differentials in this chain complex are the $\mathbb{Z}{G}$-linear maps given by

[TABLE]

where $\varphi\colon F\twoheadrightarrow G$ is the presenting homomorphism for our group.

3.2. Echelon presentations

We now introduce a special type of group presentations which will play an important role in the sequel.

Definition 3.1**.**

Let $G$ be a group with a finite presentation $P=\langle\mathbf{x}\mid\mathbf{w}\rangle$, where $\mathbf{x}=\{x_{1},\dots,x_{n}\}$ and $\mathbf{w}=\{w_{1},\dots,w_{m}\}$. We say $P$ is an echelon presentation if the augmented Fox Jacobian matrix $(\varepsilon_{i}(w_{k}))$ is in row-echelon form.

Let $K_{G}$ be the $2$-complex associated to the above presentation for $G$. Suppose the pivot elements of the $m\times n$ matrix $(\varepsilon_{i}(w_{k}))$ are in position $\{i_{1},\dots,i_{d}\}$, and let $b=n-d$. Since this matrix is in row-echelon form, the vector space $H_{1}(K_{G};\mathbb{Q})=\mathbb{Q}^{b}$ has basis $\mathbf{y}=\{y_{1},\dots,y_{b}\}$, where $y_{j}=e^{1}_{i_{d+j}}$ for $1\leq j\leq b$. Furthermore, the vector space $H_{2}(K_{G};\mathbb{Q})=\mathbb{Q}^{m-d}$ has basis $\{e^{2}_{d+1},\dots,e^{2}_{m}\}$. We will choose as basis for $H^{1}(K_{G};\mathbb{Q})$ the set $\{u_{1},\dots,u_{b}\}$, where $u_{i}$ is the Kronecker dual to $y_{i}$.

Remark 3.2**.**

Suppose $G$ admits a commutator-relators presentation of the form $P=F/R$, with $R\subset[F,F]$. Then the augmented Fox Jacobian matrix $(\varepsilon_{i}(r_{k}))$ is the zero matrix, and thus the presentation $P$ is an echelon presentation. In this case, the integer (co)homology groups of $K_{G}$ are torsion-free, and so the aforementioned choices of bases work for integer (co)homology, as well.

More generally, the next proposition shows that for any finitely presented group, we can construct a group with an echelon presentation such that the cohomology groups of the corresponding presentation $2$-complexes are isomorphic.

Proposition 3.3**.**

Let $G$ be a finitely presented group. There exists then a group $G_{e}$ with echelon presentation, and a map $f\colon K_{G_{e}}\to K_{G}$ between the respective presentation $2$-complexes such that the induced homomorphism on fundamental groups, $\rho=f_{\sharp}\colon G_{e}\twoheadrightarrow G$, is surjective, and the induced homomorphism in cohomology, $f^{*}\colon H^{*}(K_{G};\mathbb{Z})\to H^{*}(K_{G_{e}};\mathbb{Z})$, in an isomorphism.

Proof.

Suppose $G$ has presentation $\langle x_{1},\dots,x_{n}\mid r_{1},\dots,r_{m}\rangle$. As in the above discussion, the matrix of the boundary map $d_{2}^{*}\colon C^{1}(K_{G};\mathbb{Z})\to C^{2}(K_{G};\mathbb{Z})$ is the transpose of the $m\times n$ Jacobian matrix $(\varepsilon_{i}(r_{k}))$. By Gaussian elimination over $\mathbb{Z}$, there exists a matrix $C=(c_{l,k})\in\operatorname{GL}(m;\mathbb{Z})$ such that $C\cdot d_{2}^{*}$ is in row-echelon form (also known as Hermite normal form). We define a new group,

[TABLE]

by setting $w_{k}=r_{1}^{c_{1,k}}r_{2}^{c_{2,k}}\cdots r_{m}^{c_{m,k}}$ for $1\leq k\leq m$.

Let $h\colon K^{(1)}_{G_{e}}\to K^{(1)}_{G}$ be the homeomorphism between the $1$-skeleta of the respective $2$-complexes obtained by matching $1$-cells. If $\psi_{k}\colon S^{1}\to K^{(1)}_{G_{e}}$ denotes the attaching map of the $2$-cell in $K_{G_{e}}$ corresponding to the relator $w_{k}$, then by construction $h\circ\psi_{k}$ is null-homotopic in $K_{G}$. Thus, $h$ extends to a cellular map $f\colon K_{G_{e}}\to K_{G}$. Clearly, the induced homomorphism $\rho=f_{\sharp}\colon G_{e}\to G$ is surjective. Furthermore, the map $f$ induces a chain map between the respective cellular chain complexes, $f_{*}\colon C_{*}(K_{G_{e}};\mathbb{Z})\to C_{*}(K_{G};\mathbb{Z})$, with $f_{2}$ given by the matrix $C$. It is now straightforward to see that the map $f$ induces an isomorphism in homology, and thus, by the Universal Coefficients theorem, an isomorphism in cohomology, too. ∎

Note that the group $G_{e}$ constructed above depends on the given (finite) presentation for $G$, not just on the isomorphism type of $G$. On the other hand, if $G$ is a commutator-relators group, then, by Remark 3.2, the group $G_{e}$ is isomorphic to $G$.

3.3. A transferred basis

Once again, let $G$ be a group admitting a finite presentation $\langle\mathbf{x}\mid\mathbf{r}\rangle$, where $\mathbf{x}=\{x_{1},\dots,x_{n}\}$ and $\mathbf{r}=\{r_{1},\dots,r_{m}\}$, and let $K_{G}$ be the corresponding presentation $2$-complex. Using an echelon approximation for the given presentation, we describe now convenient bases for the $\mathbb{Q}$-vector spaces $H^{1}(K_{G},\mathbb{Q})$ and $H^{2}(K_{G},\mathbb{Q})$, which will be used extensively in the next two sections.

By Proposition 3.3, there exists a group $G_{e}$ with echelon presentation $\langle\mathbf{x}\mid\mathbf{w}\rangle$, where $\mathbf{w}=\{w_{1},\dots,w_{m}\}$, and a map $f\colon K_{G_{e}}\to K_{G}$ inducing an isomorphism in (co)homology. As in §3.2, we may choose a basis $\mathbf{y}=\{y_{1},\dots,y_{b}\}$ for the $\mathbb{Q}$-vector space $H_{1}(K_{G};\mathbb{Q})\cong H_{1}(K_{G_{e}};\mathbb{Q})$; let $\{u_{1},\dots,u_{b}\}$ be the dual basis for $H^{1}(K_{G};\mathbb{Q})\cong H^{1}(K_{G_{e}};\mathbb{Q})$. We also choose a basis $\{z_{1},\dots,z_{m}\}$ for $C_{2}(K_{G};\mathbb{Q})$ and a basis $\{e^{2}_{1},\dots,e^{2}_{m}\}$ for $C_{2}(K_{G_{e}};\mathbb{Q})$ corresponding to $\{1\otimes\tilde{e}_{1}^{2},\dots,1\otimes\tilde{e}_{m}^{2}\}$. Finally, if we set

[TABLE]

then $\{\gamma_{1},\dots\gamma_{m}\}$ is another basis for $C_{2}(K_{G};\mathbb{Q})$. Furthermore, $\{e_{d+1}^{2},\dots,e_{m}^{2}\}$ is a basis for $H_{2}(K_{G_{e}};\mathbb{Q})$ and $\{\gamma_{d+1},\dots,\gamma_{m}\}$ is a basis for $H_{2}(K_{G};\mathbb{Q})$. Thus, $H^{2}(K_{G};\mathbb{Q})$ has dual basis $\{\beta_{d+1},\dots,\beta_{m}\}$.

4. Group presentations and (co)homology

We compute in this section the cup-product in the cohomology ring of the $2$-complex of a finitely presented group in terms of the Magnus expansion associated to the presentation.

4.1. A chain transformation

We start by reviewing the classical bar construction. Let $G$ be a discrete group, and let $B_{*}(G)$ be the normalized bar resolution (see e.g. [4, 10]), where $B_{p}(G)$ is the free left $\mathbb{Z}G$-module on generators $[g_{1}|\dots|g_{p}]$, with $g_{i}\in G$ and $g_{i}\neq 1$, and $B_{0}(G)=\mathbb{Z}{G}$ is free on one generator, $[\ ]$. The boundary operators are $G$-module homomorphisms, $\delta_{p}\colon B_{p}(G)\rightarrow B_{p-1}(G)$, defined by

[TABLE]

In particular, $\delta_{1}[g]=(g-1)[\ ]$ and $\delta_{2}[g_{1}|g_{2}]=g_{1}[g_{2}]-[g_{1}g_{2}]+[g_{1}]$. Let $\varepsilon\colon B_{0}(G)\to\mathbb{Z}$ be the augmentation map. We then have a free resolution $B_{*}(G)\xrightarrow{\varepsilon}\mathbb{Z}$ of the trivial $G$-module $\mathbb{Z}$.

We view here $\mathbb{Z}$ as a right $\mathbb{Z}{G}$-module, with action induced by the augmentation map. An element of the cochain group $B^{p}(G)=\operatorname{Hom}_{\mathbb{Z}{G}}(B_{p}(G),\mathbb{Z})$ may be viewed as a set function $u\colon G^{p}\to\mathbb{Z}$ satisfying the normalization condition $u(g_{1},\dots,g_{p})=0$ if some $g_{i}=1$. The cup-product of two $1$-dimensional classes $u,u^{\prime}\in H^{1}(G;\mathbb{Z})\cong\operatorname{Hom}(G,\mathbb{Z})$ is given by

[TABLE]

For future use, we record a result due to Fenn and Sjerve ([10, Thm.  2.1 and p. 327]).

Lemma 4.1** ([10]).**

There exists a chain transformation $T\colon C_{*}(\widetilde{K_{G}})\rightarrow B_{*}(G)$ of augmented chain complexes,

[TABLE]

defined by $T_{0}(\lambda):=\lambda[~{}]$,

[TABLE]

where $\tau_{1}\colon B_{1}(G)\to B_{2}(G)$ is the homomorphism defined by

[TABLE]

for all $g,g_{1}\in G$.

4.2. Cup products for echelon presentations

Now let $G$ be a group with echelon presentation $G=\langle\mathbf{x}\mid\mathbf{w}\rangle$, where $\mathbf{x}=\{x_{1},\dots,x_{n}\}$ and $\mathbf{w}=\{w_{1},\dots,w_{m}\}$, as in Definition 3.1. We let $B_{*}(G;\mathbb{Q})=\mathbb{Q}\otimes B_{*}(G)$ and $B^{*}(G;\mathbb{Q})=\mathbb{Q}\otimes B^{*}(G)$.

Lemma 4.2**.**

For each basis element $u_{i}\in H^{1}(K_{G};\mathbb{Q})\cong H^{1}(G;\mathbb{Q})$ as above, and each $r\in F$, we have that

[TABLE]

where $(a_{i,s})$ is the $b\times n$ matrix for the projection map $\pi\colon F_{\mathbb{Q}}\to G_{\mathbb{Q}}$.

Proof.

If $r\in F$, then $\varphi(r)\in G$ and $[\varphi(r)]\in B_{1}(G)$. Hence,

[TABLE]

Since $H^{1}(G;\mathbb{Q})\cong B^{1}(G;\mathbb{Q})\cong\operatorname{Hom}(G,\mathbb{Q})$, we may view $u_{i}$ as a group homomorphism. This yields the first equality in (22). Since $\pi(x_{s})=\sum_{j=1}^{b}a_{i,s}y_{i}$ and $u_{i}=y_{i}^{*}$, the second equality follows. The last equality follows from Lemma 2.5. ∎

Theorem 4.3**.**

Let $G$ be a group with echelon presentation $G=\langle\mathbf{x}\mid\mathbf{w}\rangle$. The cup-product map $\mu_{K_{G}}\colon H^{1}(K_{G};\mathbb{Q})\wedge H^{1}(K_{G};\mathbb{Q})\rightarrow H^{2}(K_{G};\mathbb{Q})$ is given by $(u_{i}\cup u_{j},e^{2}_{k})=\kappa(w_{k})_{i,j}$, for $1\leq i,j\leq b$ and $d+1\leq k\leq m$, where $\kappa$ is the Magnus expansion of $G$.

Proof.

Let us write the Fox derivative $\partial_{t}(w_{k})$ as a finite sum,

[TABLE]

for $1\leq t\leq n$, and $1\leq k\leq m$. We then have

[TABLE]

The chain transformation $T\colon C_{*}(\widetilde{K_{G}};\mathbb{Q})\rightarrow B_{*}(G;\mathbb{Q})$ induces an isomorphism on first cohomology, $T^{*}\colon H^{1}(G;\mathbb{Q})\rightarrow H^{1}(K_{G};\mathbb{Q})$. Let us view $u_{i}$ and $u_{j}$ as elements in $H^{1}(G;\mathbb{Q})$. We then have

[TABLE]

and this completes the proof. ∎

4.3. Cup products for finite presentations

Let $G$ be a group with a finite presentation $\langle\mathbf{x}\mid\mathbf{r}\rangle$. By Proposition 3.3, there exists a group $G_{e}$ with echelon presentation $\langle\mathbf{x}\mid\mathbf{w}\rangle$, and a map $f\colon K_{G_{e}}\to K_{G}$ inducing an isomorphism in cohomology. Using the bases for $H^{*}(K_{G};\mathbb{Q})$ transferred from suitable bases for $H^{*}(K_{G_{e}};\mathbb{Q})$ as in §3.3, we obtain an explicit formula for computing cup-products in the rational cohomology of the presentation $2$-complex $K_{G}$.

Theorem 4.4**.**

In the aforementioned bases for $H^{*}(K_{G};\mathbb{Q})$, the cup-product map $\mu_{K_{G}}\colon H^{1}(K_{G};\mathbb{Q})\wedge H^{1}(K_{G};\mathbb{Q})\rightarrow H^{2}(K_{G};\mathbb{Q})$ is given by

[TABLE]

That is, $(u_{i}\cup u_{j},\gamma_{k})=\kappa(w_{k})_{i,j}$, for all $1\leq i,j\leq b$.

Proof.

As in the discussion from §3.3, the elements $\gamma_{k}=f_{*}(w_{k})$ with $d<k\leq m$ form a basis for $H_{2}(K_{G};\mathbb{Q})$. Hence,

[TABLE]

The claim follows. ∎

Let us consider now in more detail the case when the group $G$ is a commutator-relators group. In that case, as noted in §2.3, the Magnus expansion $\kappa=\kappa_{G}$ coincides with the classical Magnus expansion $M$. Furthermore, by Remark 3.2 both $H_{*}(K_{G};\mathbb{Z})$ and $H^{*}(K_{G};\mathbb{Z})$ are torsion-free, and the aforementioned rational bases are also integral bases for these free $\mathbb{Z}$-modules. Moreover, we may take $G_{e}=G$, and note that all the arguments from this section work over $\mathbb{Z}$ in this case. Using these observations, and the fact that $M(r_{k})_{i,j}=\varepsilon_{ij}(r_{k})$, we recover as a corollary the following result of Fenn and Sjerve [10].

Corollary 4.5** ([10], Thm. 2.4).**

For a commutator-relators group $G=\langle\mathbf{x}\mid\mathbf{r}\rangle$, the cup-product map $\mu_{K}\colon H^{1}(K_{G};\mathbb{Z})\wedge H^{1}(K_{G};\mathbb{Z})\rightarrow H^{2}(K_{G};\mathbb{Z})$ is given by $(u_{i}\cup u_{j},e_{k}^{2})=\varepsilon_{ij}(r_{k})$, for $1\leq i,j\leq n$ and $1\leq k\leq m$.

5. A presentation for the holonomy Lie algebra

In this section, we give presentations for the holonomy Lie algebra of a finitely presented group, and for the solvable quotients of this Lie algebra. In the process, we complete the proof of Theorem 1.2 from the Introduction.

5.1. The holonomy Lie algebra of a group

We start by reviewing the construction of the holonomy Lie algebra of a finitely generated group $G$, following [6, 13, 24, 31, 41]. Set

[TABLE]

where $\mu^{\vee}_{G}$ is the dual to the cup-product map $\mu_{G}\colon H^{1}(G;\mathbb{Q})\wedge H^{1}(G;\mathbb{Q})\to H^{2}(G;\mathbb{Q})$. If $\varphi\colon G_{1}\to G_{2}$ is a group homomorphism, then the induced homomorphism in cohomology, $\varphi^{*}\colon H^{1}(G_{2};\mathbb{Q})\to H^{1}(G_{1};\mathbb{Q})$, yields a morphism of graded Lie algebras, $\mathfrak{h}(\varphi)\colon\mathfrak{h}(G_{1})\to\mathfrak{h}(G_{2})$. Moreover, if $\varphi$ is surjective, then $\mathfrak{h}(\varphi)$ is also surjective.

In the definition of the holonomy Lie algebra of $G$, we used the cohomology ring of a classifying space $K(G,1)$. More generally, if $X$ is a connected space with $b_{1}(X)<\infty$, we may define its holonomy Lie algebra as $\mathfrak{h}(X)=\mathfrak{lie}(H_{1}(X;\mathbb{Q}))/\langle\operatorname{im}\mu^{\vee}_{X}\rangle$. As above, a continuous map $f\colon X\to Y$ induces a morphism $\mathfrak{h}(f)\colon\mathfrak{h}(X)\to\mathfrak{h}(Y)$; moreover, if $f\simeq g$, then $\mathfrak{h}(f)=\mathfrak{h}(g)$. The proof of the next lemma is straightforward.

Lemma 5.1**.**

Let $f\colon X\to Y$ be a map between connected spaces with finite first Betti numbers. Suppose $f$ induces isomorphisms in rational cohomology in degrees $1$ and $2$. Then the map $\mathfrak{h}(f)\colon\mathfrak{h}(X)\to\mathfrak{h}(Y)$ is an isomorphism.

In definition (25), we may replace the classifying space $K(G,1)$ used to compute group cohomology by any other connected CW-complex with the same fundamental group. The next lemma, which slightly improves on a result from [31, 41], makes this more precise.

Lemma 5.2**.**

Let $G$ be a finitely generated group, and let $X$ be a connected CW-complex with $\pi_{1}(X)=G$. There is then a natural isomorphism \mathfrak{h}(X)\xrightarrow{\,\smash{\raisebox{-2.58334pt}{\scriptstyle\simeq}}\,}\mathfrak{h}(G).

Proof.

We may construct a classifying space for the group $G$ by attaching cells of dimension $3$ and higher to the space $X$. The inclusion map, $j=j_{X}\colon X\to K(G,1)$, induces a map on cohomology rings, $j^{*}\colon H^{*}(K(G,1);\mathbb{Q})\to H^{*}(X;\mathbb{Q})$, which is an isomorphism in degree $1$ and an injection in degree $2$. In particular, $b_{1}(X)=b_{1}(G)<\infty$. Furthermore, $j^{2}$ restricts to an isomorphism from $\operatorname{im}(\mu_{G})$ to $\operatorname{im}(\mu_{X})$. Taking duals, we obtain the following diagram.

[TABLE]

Hence, the isomorphism $j_{*}\otimes j_{*}$ identifies $\operatorname{im}(\mu^{\vee}_{X})$ with $\operatorname{im}(\mu^{\vee}_{G})$. Thus, the extension to free Lie algebras of the isomorphism $j_{*}\colon H_{1}(X;\mathbb{Q})\to H_{1}(G;\mathbb{Q})$ factors through an isomorphism $\mathfrak{h}(j)\colon\mathfrak{h}(X)\to\mathfrak{h}(G)$.

To show that this isomorphism is natural, let $f\colon X\to Y$ be a map of pointed, connected CW-complexes with finitely generated fundamental groups, and let $g\colon K(\pi_{1}(X),1)\to K(\pi_{1}(Y),1)$ be the map (unique up to homotopy) induced by the homomorphism $f_{\#}\colon\pi_{1}(X)\to\pi_{1}(Y)$. Then $g\circ j_{X}\simeq j_{Y}\circ f$, and thus $\mathfrak{h}(g)\circ\mathfrak{h}(j_{X})=\mathfrak{h}(j_{Y})\circ\mathfrak{h}(f)$. ∎

Putting together the previous two lemmas, we obtain the following corollary.

Corollary 5.3**.**

Let $G_{1}$ and $G_{2}$ be two finitely generated groups, with presentation $2$-complexes $K_{1}$ and $K_{2}$. Let $f\colon K_{1}\to K_{2}$ be a cellular map, and let $\varphi=f_{\sharp}\colon G_{1}\to G_{2}$ be the induced homomorphism. If $f^{*}\colon H^{*}(K_{2},\mathbb{Q})\to H^{*}(K_{1},\mathbb{Q})$ is an isomorphism, then $\mathfrak{h}(\varphi)\colon\mathfrak{h}(G_{1})\to\mathfrak{h}(G_{2})$ is also an isomorphism.

Next, we show that, if need be, the group $G$ we started with may be replaced by a finitely presented group with the same holonomy Lie algebra.

Proposition 5.4**.**

Let $G$ be a finitely generated group. There exists then a finitely presented group $G_{f}$ and a homomorphism $G_{f}\to G$ inducing an isomorphism \mathfrak{h}(G_{f})\xrightarrow{\,\smash{\raisebox{-2.58334pt}{\scriptstyle\simeq}}\,}\mathfrak{h}(G).

Proof.

Let $X$ be a connected CW-complex with $\pi_{1}(X)=G$. Since $G$ is finitely generated, we may assume $X$ has finitely many $1$-cells. The proof of Proposition 4.1 from [34] shows that there exists a connected, finite subcomplex $Z$ of $X$ such that the inclusion $Z\to X$ induces isomorphisms $H_{1}(Z;\mathbb{Q})\cong H_{1}(X;\mathbb{Q})$ and $\operatorname{im}(\mu^{\vee}_{Z})\cong\operatorname{im}(\mu^{\vee}_{X})$. Consequently, $\mathfrak{h}(Z)\cong\mathfrak{h}(X)$. Letting $G_{f}=\pi_{1}(Z)$, the claim follows from Lemma 5.2. ∎

5.2. Magnus expansion and holonomy

Let $G$ be a group admitting a finite presentation, $P=\langle\mathbf{x}\mid\mathbf{r}\rangle$. As shown in Proposition 3.3, there exists a group $G_{e}$ with echelon presentation $P_{e}=\langle\mathbf{x}\mid\mathbf{w}\rangle$, and a map $f\colon K_{G_{e}}\to K_{G}$ inducing an isomorphism in cohomology and an epimorphism $\rho\colon G_{e}\twoheadrightarrow G$ on fundamental groups. By Corollary 5.3, the induced map between the respective holonomy Lie algebras, \mathfrak{h}(\rho)\colon\mathfrak{h}(G_{e})\xrightarrow{\,\smash{\raisebox{-2.58334pt}{\scriptstyle\simeq}}\,}\mathfrak{h}(G), is an isomorphism.

So let us consider a group $G=F/R$ admitting an echelon presentation $P=\langle\mathbf{x}\mid\mathbf{w}\rangle$, where $\mathbf{x}=\{x_{1},\dots,x_{n}\}$ and $\mathbf{w}=\{w_{1},\dots,w_{m}\}$. We now give a more explicit presentation for the holonomy Lie algebra $\mathfrak{h}(G)$.

Let $\partial_{i}(w_{k})\in\mathbb{Z}{F}$ be the Fox derivatives of the relations, and let $\varepsilon_{i}(w_{k})=\varepsilon(\partial_{i}(w_{k}))\in\mathbb{Z}$ be their augmentations. Recall from §4.2 that we can choose a basis $\mathbf{y}=\{y_{1},\dots,y_{b}\}$ for $H_{1}(K_{P};\mathbb{Q})$ and a basis $\{e^{2}_{d+1},\dots,e^{2}_{m}\}$ for $H_{2}(K_{P};\mathbb{Q})$, where $d$ is the rank of Jacobian matrix $J_{P}=(\varepsilon_{i}(w_{k}))$, viewed as an $m\times n$ matrix over $\mathbb{Q}$. Let $\mathfrak{lie}(\mathbf{y})$ be the free Lie algebra over $\mathbb{Q}$ generated by $\mathbf{y}$ in degree $1$. Recall that $\kappa_{2}$ is the degree $2$ part of the Magnus expansion of $G$ given explicitly in (14). Thus, we can identify $\kappa_{2}(w_{k})$ with $\sum_{i<j}\kappa(w_{k})_{i,j}[y_{i},y_{j}]$ in $\mathfrak{lie}(\mathbf{y})$ for $d+1\leq k\leq m$.

Theorem 5.5**.**

Let $G$ be a group admitting an echelon presentation $P=\langle\mathbf{x}\mid\mathbf{w}\rangle$. Then there exists an isomorphism of graded Lie algebras

[TABLE]

Proof.

Combining Theorem 4.3 with the fact that $(u_{i}\wedge u_{j},\mu^{\vee}(e^{2}_{k}))=(\mu(u_{i}\wedge u_{j}),e^{2}_{k})$, we see that the dual cup-product map, $\mu^{\vee}\colon H_{2}(K_{P};\mathbb{Q})\rightarrow H_{1}(K_{P};\mathbb{Q})\wedge H_{1}(K_{P};\mathbb{Q})$, is given by

[TABLE]

Hence, the following diagram commutes,

[TABLE]

Using now the identification of $\kappa_{2}(w_{k})$ and $\sum_{i<j}\kappa(w_{k})_{i,j}[y_{i},y_{j}]$ as elements of $\mathfrak{lie}(\mathbf{y})$, the definition of the holonomy Lie algebra, and the fact that $\mathfrak{h}(G)\cong\mathfrak{h}(K_{P})$, we arrive at the desired conclusion. ∎

Corollary 5.6**.**

The universal enveloping algebra of $\mathfrak{h}(G)$ has presentation

[TABLE]

If $G=\langle\mathbf{x}\mid\mathbf{r}\rangle$ is a commutator-relators group, then the group $H_{1}(K_{G};\mathbb{Z})$ is torsion-free, and thus the integer holonomy Lie ring $\mathfrak{h}(G;\mathbb{Z})$ can be defined as in (25), using integer (co)homology, instead, see [24, 31] for details. Furthermore, as in §2.2, for each $r\in[F,F]$, the primitive element $M_{2}(r)\in\widehat{T}_{2}(F_{\operatorname{ab}})$ corresponds to the element $\sum_{i<j}\varepsilon_{i,j}(r)[x_{i},x_{j}]$ from the degree $2$ piece of the free Lie ring $\mathfrak{lie}_{\mathbb{Z}}(\mathbf{x})$. Using this observation and Corollary 4.5, we recover a result from [31].

Corollary 5.7** ([31], Prop. 7.2).**

If $G=\langle x_{1},\dots,x_{n}\mid\mathbf{r}\rangle$ is a commutator-relators group, then

[TABLE]

Proposition 5.8**.**

For every quadratic Lie algebra $\mathfrak{g}$ over $\mathbb{Q}$, there exists a commutator-relators group $G_{c}$ such that $\mathfrak{h}(G_{c})=\mathfrak{g}$.

Proof.

We use an approach similar to the proof of [35, Prop. 6.2]. By assumption, we may write $\mathfrak{g}=\mathfrak{lie}(\mathbf{x})/\mathfrak{a}$, where $\mathbf{x}=\{x_{1},\dots,x_{n}\}$ and $\mathfrak{a}$ is an ideal generated by elements of the form $\ell_{k}=\sum_{1\leq i<j\leq n}c_{ijk}[x_{i},x_{j}]$ for $1\leq k\leq m$, and where the coefficients $c_{ijk}$ are in $\mathbb{Q}$. Clearing denominators, we may assume all $c_{ijk}$ are integers. We can then define words $r_{k}=\prod_{1\leq i<j\leq n}[x_{i},x_{j}]^{c_{ijk}}$ in the free group generated by $\mathbf{x}$, and set $G_{c}=\langle\mathbf{x}\mid r_{1},\dots,r_{m}\rangle$. Clearly, $\varepsilon_{i,j}(r_{k})=c_{ijk}$. The desired conclusion follows from Corollary 5.7. ∎

Corollary 5.9**.**

For every finitely generated group $G$, there exists a commutator-relators group $G_{c}$ such that $\mathfrak{h}(G_{c})=\mathfrak{h}(G)$.

Proof.

From Proposition 5.4, the holonomy Lie algebra $\mathfrak{h}(G)$ has a quadratic presentation. Letting $\mathfrak{g}=\mathfrak{h}(G)$ and applying Proposition 5.8 yields the desired conclusion. ∎

5.3. Solvable quotients of holonomy Lie algebras

The next lemma follows straight from the definitions, using the standard isomorphism theorems.

Lemma 5.10**.**

Let $\mathfrak{g}=\mathfrak{lie}(V)/\mathfrak{r}$ be a finitely generated Lie algebra. Then $\mathfrak{g}/\mathfrak{g}^{(i)}\cong\mathfrak{lie}(V)/(\mathfrak{r}+\mathfrak{lie}(V)^{(i)})$. Furthermore, if $\mathfrak{r}$ is a homogeneous ideal, then this is an isomorphism of graded Lie algebras.

The next result sharpens and extends the first part of Theorem 7.3 from [31].

Theorem 5.11**.**

Let $G=\langle\mathbf{x}\mid\mathbf{r}\rangle$ be a finitely presented group, and set $\mathfrak{h}=\mathfrak{h}(G)$. Let $\mathbf{y}=\{y_{1},\dots,y_{b}\}$ be a basis of $H_{1}(G;\mathbb{Q})$. Then, for each $i\geq 2$,

[TABLE]

where $b=b_{1}(G)$ and $w_{k}$ is defined in (16).

Proof.

By Theorem 5.5, the holonomy Lie algebra $\mathfrak{h}$ is isomorphic to the quotient of the free Lie algebra $\mathfrak{lie}(\mathbf{y})$ by the ideal generated by $\kappa_{2}(w_{n-b+1}),\dots,\kappa_{2}(w_{m})$. The claim follows from Lemma 5.10. ∎

Using Corollary 5.7, we obtain the following consequence.

Corollary 5.12**.**

Let $G=\langle x_{1},\dots,x_{n}\mid r_{1},\dots,r_{m}\rangle$ be a commutator-relators group, and let $\mathfrak{h}=\mathfrak{h}(G)$. Then, for each $i\geq 2$, the Lie algebra $\mathfrak{h}/\mathfrak{h}^{(i)}$ is isomorphic to the quotient of the free Lie algebra $\mathfrak{lie}(\mathbf{x})$ by the sum of the ideals $M_{2}(r_{1}),\dots,M_{2}(r_{m})$, and $\mathfrak{lie}^{(i)}(\mathbf{x})$.

6. Lower central series and the holonomy Lie algebra

6.1. Lower central series

Let $G$ be a finitely generated group, and let $\{\Gamma_{k}G\}_{k\geq 1}$ be its lower central series (LCS). The LCS quotients of $G$ are finitely generated abelian groups. Taking the direct sum of these groups, we obtain a graded Lie ring over $\mathbb{Z}$,

[TABLE]

The Lie bracket $[x,y]$ on $\operatorname{gr}(G;\mathbb{Z})$ is induced from the group commutator, $[x,y]=xyx^{-1}y^{-1}$. More precisely, if $x\in\Gamma_{r}G$ and $y\in\Gamma_{s}G$, then $[x+\Gamma_{r+1}G,y+\Gamma_{s+1}G]=xyx^{-1}y^{-1}+\Gamma_{r+s+1}G$. The Lie algebra $\operatorname{gr}(G;\mathbb{Q})=\operatorname{gr}(G;\mathbb{Z})\otimes\mathbb{Q}$ is called the associated graded Lie algebra (over $\mathbb{Q}$) of the group $G$. For simplicity, we will usually drop the $\mathbb{Q}$-coefficients, and simply write it as $\operatorname{gr}(G)$.

The group $G$ is said to be nilpotent of class $\leq k$ if $\Gamma_{k+1}G=\{1\}$. For each $k\geq 2$, the factor group $G/\Gamma_{k}G$ is the maximal $(k-1)$-step nilpotent quotient of $G$. The canonical projection $G\to G/\Gamma_{k}G$ induces an epimorphism $\operatorname{gr}(G)\to\operatorname{gr}(G/\Gamma_{k}G)$, which is an isomorphism in degrees $s<k$. We refer to Lazard [18] and Magnus et al. [23] for more details..

6.2. A comparison map

Once again, let $G$ be a finitely generated group. Although the next lemma is known, we provide a proof, both for the sake of completeness, and for later use.

Lemma 6.1** ([24, 31]).**

There exists a natural epimorphism of graded $\mathbb{Q}$-Lie algebras,

[TABLE]

inducing isomorphisms in degrees $1$ and $2$.

Proof.

As first noted by Sullivan [45] in a particular case, and then proved by Lambe [17] in general, there is a natural exact sequence

[TABLE]

where $\beta$ is the dual of Lie bracket product. Consequently, $\operatorname{im}(\mu^{\vee}_{G})=\ker(\beta^{\vee})$.

Recall now that the associated graded Lie algebra $\operatorname{gr}(G)$ is generated by its degree $1$ piece, $H_{1}(G;\mathbb{Q})\cong\operatorname{gr}_{1}(G)$. Hence, there is a natural epimorphism of graded $\mathbb{Q}$-Lie algebras,

[TABLE]

restricting to the identity in degree $1$, and to the Lie bracket map $[\,,\,]\colon\bigwedge^{2}\operatorname{gr}_{1}(G)\rightarrow\operatorname{gr}_{2}(G)$ in degree $2$. By the above observation, the kernel of this map coincides with the image of $\mu^{\vee}_{G}$. Thus, $\varphi_{G}$ factors through a morphism $\Phi_{G}\colon\mathfrak{h}(G)\to\operatorname{gr}(G)$, which enjoys all the claimed properties. ∎

6.3. Nilpotent and derived quotients

As a quick application, let us compare the holonomy Lie algebra of a group to the holonomy Lie algebras of its nilpotent quotients and derived quotients.

Proposition 6.2**.**

Let $G$ be a finitely generated group. Then

[TABLE]

In particular, the holonomy Lie algebra of $G$ depends only on the second nilpotent quotient, $G/\Gamma_{3}G$.

Proof.

The case $k=2$ is trivial, so let us assume $k\geq 3$. By a previous remark, the projection $G\to G/\Gamma_{k}G$ induces an isomorphism $\operatorname{gr}_{2}(G)\to\operatorname{gr}_{2}(G/\Gamma_{k}G)$. Furthermore, $H_{1}(G;\mathbb{Q})\cong H_{1}(G/\Gamma_{k}G)$. Using now the dual of the exact sequence (30), we see that $\operatorname{im}(\mu^{\vee}_{G})=\operatorname{im}(\mu^{\vee}_{G/\Gamma_{k}G})$. The desired conclusion follows. ∎

Proposition 6.3**.**

The holonomy Lie algebras of the derived quotients of $G$ are given by

[TABLE]

Proof.

For $i=1$, the statement trivially holds, so we may as well assume $i\geq 2$. It is readily proved by induction that $G^{(i)}\subseteq\Gamma_{2^{i}}(G)$. Hence, the projections

[TABLE]

yield natural projections $\mathfrak{h}(G)\twoheadrightarrow\mathfrak{h}(G/G^{(i)})\twoheadrightarrow\mathfrak{h}(G/\Gamma_{2^{i}}G)=\mathfrak{h}(G)$. By Proposition 6.2, the composition of these projections is an isomorphism of Lie algebras. Therefore, the surjection $\mathfrak{h}(G)\twoheadrightarrow\mathfrak{h}(G/G^{(i)})$ is an isomorphism. ∎

An analogous result holds for associated graded Lie algebras, albeit in somewhat weaker form. The quotient map, $q_{i}\colon G\twoheadrightarrow G/G^{(i)}$, induces a surjective morphism between associated graded Lie algebras. Plainly, this morphism is the canonical identification in degree $1$. In fact, more is true.

Lemma 6.4**.**

For each $i\geq 2$ and each $k\leq 2^{i}-1$, the map $\operatorname{gr}(q_{i})\colon\operatorname{gr}_{k}(G)\twoheadrightarrow\operatorname{gr}_{k}(G/G^{(i)})$ is an isomorphism.

Proof.

Taking associated graded Lie algebras in sequence (32) yields epimorphisms

[TABLE]

By a remark we made at the end of §6.1, the composition of these maps is an isomorphism in degrees $k<2^{i}$. The conclusion follows at once. ∎

The next result distills the statements of Theorem 9.3 and Corollary 9.5 from [41], in a form needed here; this result sharpens and extends Theorem 4.2 from [31].

Theorem 6.5** ([41]).**

Let $G$ be a finitely generated group. For each $i\geq 2$, the quotient map $G\twoheadrightarrow G/G^{(i)}$ induces a natural epimorphism of graded Lie algebras, $\operatorname{gr}(G)/\operatorname{gr}(G)^{(i)}\twoheadrightarrow\operatorname{gr}(G/G^{(i)})$. Moreover, if $G$ is a $1$-formal group, then $\mathfrak{h}(G)/\mathfrak{h}(G)^{(i)}\cong\operatorname{gr}(G/G^{(i)})$.

Combining Theorem 6.5 with Theorem 5.11, we obtain the following corollary.

Corollary 6.6**.**

Let $G=\langle\mathbf{x}\mid\mathbf{r}\rangle$ be a finitely presented, $1$-formal group. Let $\mathbf{y}=\{y_{1},\dots,y_{b}\}$ be a basis of $H_{1}(G;\mathbb{Q})$. Then, for each $i\geq 2$,

[TABLE]

where $b=b_{1}(G)$ and $w_{k}$ is defined in (16).

6.4. Graded-formality

We conclude our discussion of associated graded Lie algebras and holonomy Lie algebras by recalling a notion that will be important in the sequel. Recall from Lemma 6.1 that, for any finitely generated group $G$, there is a canonical epimorphism of graded Lie algebras, $\Phi_{G}\colon\mathfrak{h}(G)\twoheadrightarrow\operatorname{gr}(G)$. We say that the group $G$ is graded-formal (over $\mathbb{Q}$) if the map $\Phi_{G}$ is an isomorphism.

This notion was considered in various ways by Chen [6], Kohno [13], Labute [15], and Hain [11]. It was also recently studied by Lee in [19], where it was called ‘graded $1$-formality.’ Various relationships between graded-formality and other formality properties were studied in [41]. In particular, a finitely generated group $G$ is $1$-formal if and only if it is both graded-formal and filtered-formal. We give here two alternate characterizations of graded formality, which oftentimes are easier to verify.

Proposition 6.7**.**

A finitely generated group $G$ is graded-formal if and only if one of the following two conditions is satisfied.

1. (1)

The Lie algebra $\operatorname{gr}(G)$ is quadratic. 2. (2)

$\dim_{\mathbb{Q}}\mathfrak{h}_{n}(G)=\dim_{\mathbb{Q}}\operatorname{gr}_{n}(G)$, for all $n\geq 1$.

Proof.

Clearly, if the group $G$ is graded-formal, then both conditions are satisfied.

Assume now that (1) holds, that is, the graded Lie algebra $\operatorname{gr}(G)$ admits a presentation of the form $\mathfrak{lie}(V)/\langle U\rangle$, where $V$ is a finite-dimensional $\mathbb{Q}$-vector space concentrated in degree $1$ and $U$ is a $\mathbb{Q}$-vector subspace of $\mathfrak{lie}_{2}(V)$. In particular, $V=\operatorname{gr}_{1}(G)=H_{1}(G;\mathbb{Q})$. From the exact sequence (30), we see that the image of $\mu^{\vee}_{G}$ coincides with the kernel of the Lie bracket map $[\,,\,]\colon\bigwedge^{2}\operatorname{gr}_{1}(G)\rightarrow\operatorname{gr}_{2}(G)$, which can be identified with $U$. Hence, the surjection $\varphi_{G}\colon\mathfrak{lie}(V)\twoheadrightarrow\operatorname{gr}(G)$ induces an isomorphism \Phi_{G}\colon\mathfrak{h}(G)\xrightarrow{\,\smash{\raisebox{-2.58334pt}{\scriptstyle\simeq}}\,}\operatorname{gr}(G).

Finally, assume (2) holds. In general, the homomorphism $(\Phi_{G})_{n}\colon\mathfrak{h}_{n}(G)\to\operatorname{gr}_{n}(G)$ is an isomorphism for $n\leq 2$ and an epimorphism for $n\geq 3$. Our assumption, together with the fact that each $\mathbb{Q}$-vector space $\mathfrak{h}_{n}(G)$ is finite-dimensional implies that all homomorphisms $(\Phi_{G})_{n}$ are isomorphisms. Therefore, the map $\Phi_{G}\colon\mathfrak{h}(G)\twoheadrightarrow\operatorname{gr}(G)$ is an isomorphism of graded Lie algebras. ∎

7. Mildness and graded-formality

We start this section with the notion of mild (or inert) presentation of a group, due to J. Labute and D. Anick, and its relevance to the associated graded Lie algebra. We then continue with some applications to two important classes of finitely presented groups: one-relator groups and fundamental groups of link complements.

7.1. Mild presentations

Let $F$ be a free group generated by $\mathbf{x}=\{x_{1},\dots,x_{n}\}$. The weight of a word $r\in F$ is defined as $\omega(r)=\sup\{k\mid r\in\Gamma_{k}F\}$. Since $F$ is residually nilpotent, $\omega(r)$ is finite. The image of $r$ in $\operatorname{gr}_{\omega(r)}(F)$ is called the initial form of $r$, and is denoted by $\operatorname{in}(r)$.

Let $G=F/R$ be a quotient of $F$, with presentation $G=\langle\mathbf{x}\mid\mathbf{r}\rangle$, where $\mathbf{r}=\{r_{1},\dots,r_{m}\}$. Let $\operatorname{in}(\mathbf{r})$ be the ideal of the free $\mathbb{Q}$-Lie algebra $\mathfrak{lie}(\mathbf{x})$ generated by $\{\operatorname{in}(r_{1}),\dots,\operatorname{in}(r_{m})\}$. Clearly, this is a homogeneous ideal; thus, the quotient

[TABLE]

is a graded Lie algebra. As noted by Labute in [15], the ideal $\operatorname{in}(\mathbf{r})$ is contained in $\operatorname{gr}^{\widetilde{\Gamma}}(R)$, where ${\widetilde{\Gamma}}_{k}R=\Gamma_{k}{F}\cap R$ is the induced filtration on $R$. Hence, there exists an epimorphism $L(G)\twoheadrightarrow\operatorname{gr}(G)$.

Proposition 7.1**.**

Let $G$ be a commutator-relators group, and let $\mathfrak{h}(G)$ be its holonomy Lie algebra. Then the canonical projection $\Phi_{G}\colon\mathfrak{h}(G)\twoheadrightarrow\operatorname{gr}(G)$ factors through an epimorphism $\mathfrak{h}(G)\twoheadrightarrow L(G)$.

Proof.

Let $G=\langle\mathbf{x}\mid\mathbf{r}\rangle$ be a commutator-relators presentation for our group. By Corollary 5.7, the holonomy Lie algebra $\mathfrak{h}(G)$ admits a presentation of the form $\mathfrak{lie}(\mathbf{x})/\mathfrak{a}$, where $\mathfrak{a}$ is the ideal generated by the degree $2$ part of $M(r)-1$, for all $r\in\mathbf{r}$. On the other hand, $\operatorname{in}(r)$ is the smallest degree homogeneous part of $M(r)-1$. Hence, $\mathfrak{a}\subseteq\operatorname{in}(\mathbf{r})$, and this complete the proof. ∎

Following [1, 15], we say that a group $G$ is a mildly presented group (over $\mathbb{Q}$) if it admits a presentation $G=\langle\mathbf{x}\mid\mathbf{r}\rangle$ such that the quotient $\operatorname{in}(\mathbf{r})/[\operatorname{in}(\mathbf{r}),\operatorname{in}(\mathbf{r})]$, viewed as a $U(L(G))$-module via the adjoint representation of $L(G)$, is a free module on the images of $\operatorname{in}(r_{1}),\dots,\operatorname{in}(r_{m})$. As shown by Anick in [1], a presentation $G=\langle x_{1},\dots,x_{n}\mid r_{1},\dots r_{m}\rangle$ is mild if and only if

[TABLE]

Theorem 7.2** (Labute [14, 15]).**

Let $G$ be a finitely-presented group.

1. (1)

If $G$ is mildly presented, then $\operatorname{gr}(G)=L(G)$. 2. (2)

If $G$ has a single relator $r$, then $G$ is mildly presented. Moreover, the LCS ranks $\phi_{k}(G):=\operatorname{rank}\operatorname{gr}_{k}(G)$ are given by

[TABLE]

where $\mu$ is the Möbius function and $e=\omega(r)$.

Labute states this theorem over $\mathbb{Z}$, but his proof works for any commutative PID with unity. There is an example in [15] showing that the mildness condition is crucial for part (1) of the theorem to hold. We give now a much simpler example to illustrate this phenomenon.

Example 7.3**.**

Let $G=\langle x_{1},x_{2},x_{3}\mid x_{3},x_{3}[x_{1},x_{2}]\rangle$. Clearly, $G\cong\langle x_{1},x_{2}\mid[x_{1},x_{2}]\rangle$, which is a mild presentation. However, the Lie algebra $\mathfrak{lie}(x_{1},x_{2},x_{3})/{\rm ideal}(x_{3})$ is not isomorphic to $\operatorname{gr}(G)=\mathfrak{lie}(x_{1},x_{2})/{\rm ideal}([x_{1},x_{2}])$. Hence, the first presentation is not a mild.

Under different assumptions, alternative methods for computing the LCS ranks of a group $G$ can be found in [47, 43].

7.2. Mildness and graded formality

We now use Labute’s work on the associated graded Lie algebra and our presentation of the holonomy Lie algebra to give two graded-formality criteria.

Corollary 7.4**.**

Let $G$ be a group admitting a mild presentation $\langle\mathbf{x}\mid\mathbf{r}\rangle$. If $\omega(r)\leq 2$ for each $r\in\mathbf{r}$, then $G$ is graded-formal.

Proof.

By Theorem 7.2, the associated graded Lie algebra $\operatorname{gr}(H;\mathbb{Q})$ has a presentation of the form $\mathfrak{lie}(\mathbf{x})/\operatorname{in}(\mathbf{r})$, with $\operatorname{in}(\mathbf{r})$ a homogeneous ideal generated in degrees $1$ and $2$. Using the degree $1$ relations to eliminate superfluous generators, we arrive at a presentation with only quadratic relations. The desired conclusion follows from Proposition 6.7. ∎

An important sufficient condition for mildness of a presentation was given by Anick [1]. Recall that $\iota$ denotes the canonical injection from the free Lie algebra $\mathfrak{lie}(\mathbf{x})$ into $\mathbb{Q}\langle\mathbf{x}\rangle$. Fix an ordering on the set $\{\mathbf{x}\}$. The set of monomials in the homogeneous elements $\iota(\operatorname{in}(r_{1})),\dots,\iota(\operatorname{in}(r_{m}))$ inherits the lexicographic order. Let $w_{i}$ be the highest term of $\iota(\operatorname{in}(r_{i}))$ for $1\leq i\leq m$. Suppose that (i) no $w_{i}$ equals zero; (ii) no $w_{i}$ is a submonomial of any $w_{j}$ for $i\neq j$, i.e., $w_{j}=uw_{i}v$ cannot occur; and (iii) no $w_{i}$ overlaps with any $w_{j}$, i.e., $w_{i}=uv$ and $w_{j}=vw$ cannot occur unless $v=1$, or $u=w=1$. Then, the set $\{r_{1},\dots,r_{n}\}$ is mild (over $\mathbb{Q}$). We use this criterion to provide an example of a finitely-presented group $G$ which is graded-formal, but not filtered-formal.

Example 7.5**.**

Let $G$ be the group with generators $x_{1},\dots,x_{4}$ and relators $r_{1}=[x_{2},x_{3}]$, $r_{2}=[x_{1},x_{4}]$, and $r_{3}=[x_{1},x_{3}][x_{2},x_{4}]$. Ordering the generators as $x_{1}\succ x_{2}\succ x_{3}\succ x_{4}$, we find that the highest terms for $\{\iota(\operatorname{in}(r_{1})),\iota(\operatorname{in}(r_{2})),\iota(\operatorname{in}(r_{3}))\}$ are $\{x_{2}x_{3},x_{1}x_{4},x_{1}x_{3}\}$, and these words satisfy the above conditions of Anick. Thus, by Theorem 7.2, the Lie algebra $\operatorname{gr}(G)$ is the quotient of $\mathfrak{lie}(x_{1},\dots,x_{4})$ by the ideal generated by $[x_{2},x_{3}]$, $[x_{1},x_{4}]$, and $[x_{1},x_{3}]+[x_{2},x_{4}]$. Hence, $\mathfrak{h}(G)\cong\operatorname{gr}(G)$, that is, $G$ is graded-formal. On the other hand, using the Tangent Cone Theorem from [7], one can show that the group $G$ is not $1$-formal. Therefore, $G$ is not filtered-formal.

7.3. The rational Murasugi conjecture

Let $L=(L_{1},\dots,L_{n})$ be an $n$-component link in $S^{3}$. The complement of the link, $X=S^{3}\setminus\bigcup_{i=1}^{n}L_{i}$, is a a connected, $3$-dimensional manifold, which has the homotopy type of a finite, $2$-dimensional CW-complex. The link group, $G=\pi_{1}(X)$, carries crucial information about the homotopy type of $X$. For instance, if $n=1$ (i.e., the link is a knot), or if $n>1$ and $L$ is a not a split link, then the complement $X$ is a $K(G,1)$.

Let $\ell_{i,j}=\operatorname{lk}(L_{i},L_{j})$ be the linking numbers of $L$. The information coming from these numbers is conveniently encoded in a graph $\Gamma$ with vertex set $\{1,\dots,n\}$, and edges $(i,j)$ whenever $\ell_{i,j}\neq 0$. As noted in [24, 31], the holonomy Lie algebra $\mathfrak{h}(G)=\mathfrak{h}(X)$ is determined by these data:

[TABLE]

Turning now to the associated graded Lie algebra of a link group $G$, Murasugi conjectured in [29] that $\operatorname{gr}_{k}(G;\mathbb{Z})=\operatorname{gr}_{k}(F_{n-1};\mathbb{Z})$ for all $k>1$, provided that the link $L$ has $n$ components, and all the linking numbers are equal to $\pm 1$. This conjecture was proved by Massey–Traldi [26] and Labute [16], who also proved an analogous result for the Chen ranks of such links. In [46], Traldi computed the Chen groups $\operatorname{gr}_{k}(G/G^{\prime\prime};\mathbb{Z})$ for all links with connected linking graph. The next theorem, which can be viewed as a rational version of Murasugi’s conjecture, combines results of Anick [1], Berceanu–Papadima [3], and Papadima–Suciu [31].

Theorem 7.6**.**

Let $L$ be an $n$-component link in $S^{3}$, and let $G$ be the fundamental group of its complement. Assuming the linking graph $\Gamma$ is connected, the following hold.

1. (1)

The group $G$ is graded-formal, and thus, the associated graded Lie algebra $\operatorname{gr}(G)$ is isomorphic to the holonomy Lie algebra $\mathfrak{h}(G)$, with presentation given by (37). 2. (2)

There exists a graded Lie algebra isomorphism $\operatorname{gr}(G/G^{\prime\prime})\cong\mathfrak{h}(G)/\mathfrak{h}(G)^{\prime\prime}$. 3. (3)

If, furthermore, $L$ is the closure of a pure braid, then $G$ admits a mild presentation.

Proof.

The first assertion follows from Lemma 4.1 and Theorems 3.2 and 4.2 in [3], the second assertion is proved in [31, Thm. 10.1], while the last assertion follows from [1, Thm.  3.7]. ∎

We conclude this section with several examples illustrating the concepts discussed above. In each example, $L$ is a link in $S^{3}$, and $G$ is the corresponding link group. The first two examples were computed by Hain in [11] using a slightly different method.

Example 7.7**.**

Let $L$ be the Borromean rings. All the linking numbers are [math], and so $\mathfrak{h}(G)=\mathfrak{lie}(x,y,z)$. The link group $G$ has a presentation with three generators $x,y,z$ and two relators, $r_{1}=[x,[y,z]]$ and $r_{2}=[z,[y,x]]$. It is readily seen that the link $L$ passes Anick’s mildness test; hence $G$ admits a mild presentation. Thus, $\operatorname{gr}(G)=\mathfrak{lie}(x,y,z)/\operatorname{ideal}([x,[y,z]],[z,[y,x]])$, and so $G$ is not graded-formal.

Example 7.8**.**

Let $L$ be the Whitehead link. This is a $2$-component link with linking number [math]; its link group has presentation $G=\langle x,y\mid r\rangle$, where $r=x^{-1}y^{-1}xyx^{-1}yxy^{-1}xyx^{-1}y^{-1}xy^{-1}x^{-1}y$. Since $G$ has only one relator, Theorem 7.2 insures that this presentation is mild. Direct computation shows that $\operatorname{in}(r)=[x,[y,[x,y]]]$. Thus, $\operatorname{gr}(G)=\mathfrak{lie}(x,y)/\operatorname{ideal}([x,[y,[x,y]]])$, and $G$ is not graded-formal.

Example 7.9**.**

Let $L$ be the link of great circles in $S^{3}$ corresponding to the arrangement of transverse planes through the origin of $\mathbb{R}^{4}$ denoted as $\mathscr{A}(31425)$ in Matei–Suciu [28]. By construction, $L$ is the closure of a pure braid, and its linking graph is a complete graph. Thus, by Theorem 7.6, the link group $G$ is graded-formal, and admits a mild presentation. On the other hand, as noted in [7, Example 8.2], the group $G$ is not $1$-formal.

8. One-relator groups

We turn now to some other specific classes of finitely presented groups where our approach applies. We start with a well-known and much-studied class of groups in group theory.

8.1. Holonomy and graded-formality

If the group $G$ admits a finite presentation with a single relator, we saw in the previous section that $G$ is mildly presented. In fact, more can be said.

Proposition 8.1**.**

Let $G=\langle\mathbf{x}\mid r\rangle$ be a $1$-relator group.

1. (1)

If $r$ is a commutator relator, then $\mathfrak{h}(G)=\mathfrak{lie}(\mathbf{x})/{\rm ideal}(M_{2}(r))$. 2. (2)

If $r$ is not a commutator relator, then $\mathfrak{h}(G)=\mathfrak{lie}(y_{1},\dots,y_{n-1})$.

Proof.

Part (1) follows from Corollary 5.7. When $r$ is not a commutator relator, the Jacobian matrix $J_{G}=(\varepsilon(\partial_{i}r))$ has rank $1$. Part (2) then follows from Theorem 5.5. ∎

Corollary 8.2**.**

Let $G=\langle x_{1},\dots x_{n}\mid r\rangle$ be a $1$-relator group, and let $\mathfrak{h}=\mathfrak{h}(G)$. Then

[TABLE]

Proof.

Let $\mathbf{x}=\{x_{1},\dots,x_{n}\}$. By Proposition 8.1, the universal enveloping algebra $U(\mathfrak{h})$ is isomorphic to either $T(y_{1},\dots,y_{n-1})$ if $\omega(r)=1$, or to $T(\mathbf{x})/\operatorname{ideal}(M_{2}(r))$ if $\omega(r)=2$, or to $T(\mathbf{x})$ if $\omega(r)\geq 3$. The claim now follows from Theorem 7.2 and formula (35). ∎

Theorem 8.3**.**

Let $G=\langle\mathbf{x}\mid r\rangle$ be a group defined by a single relation. Then $G$ is graded-formal if and only if $\omega(r)\leq 2$.

Proof.

By Theorem 7.2, the given presentation of $G$ is mild. The weight $\omega(r)$ can also be computed as $\omega(r)=\inf\{\left|I\right|\mid M(r)_{I}\neq 0\}$. If $\omega(r)\leq 2$, then, by Proposition 8.1, we have that $\mathfrak{h}(G)\cong\operatorname{gr}(G)\cong\mathfrak{lie}(\mathbf{x})/{\rm ideal}(\operatorname{in}(r))$, and so $G$ is graded-formal.

On the other hand, if $\omega(r)\geq 3$, then $\mathfrak{h}(G)=\mathfrak{lie}(\mathbf{x})$. However, $\operatorname{gr}(G)=\mathfrak{lie}(\mathbf{x})/{\rm ideal}(\operatorname{in}(r))$. Hence, $G$ is not graded-formal. ∎

Example 8.4**.**

Let $G=\langle x_{1},x_{2}\mid r\rangle$, where $r=[x_{1},[x_{1},x_{2}]]$. Clearly, $\omega(r)=3$. Hence, $G$ is not graded-formal.

The next example shows that there exists a graded-formal group which is not filtered-formal.

Example 8.5**.**

Let $G=\langle x_{1},\dots,x_{5}\mid w\rangle$, where $w=[x_{1},x_{2}][x_{3},[x_{4},x_{5}]]$. Since $\omega(w)=2$, Theorem 8.3 implies that the group $G$ is graded-formal. On the other hand, as shown in [41], $G$ is not $1$-formal, and so $G$ is not filtered-formal.

8.2. Chen ranks

We now determine the ranks of the (rational) Chen Lie algebra associated to an arbitrary finitely presented, $1$-relator, $1$-formal group, thereby extending a result of Papadima and Suciu from [31]. By definition, the Chen ranks of a finitely generated group $G$ are the LCS ranks of its maximal metabelian quotient,

[TABLE]

The projection $\pi\colon G\twoheadrightarrow G/G^{\prime\prime}$ induces an epimorphism, $\operatorname{gr}(\pi)\colon\operatorname{gr}(G)\twoheadrightarrow\operatorname{gr}(G/G^{\prime\prime})$, which is an isomorphism in degrees $k\leq 3$. Consequently, $\phi_{k}(G)\geq\theta_{k}(G)$, with equality for $k\leq 3$. The Chen ranks were introduced and studied by K.-T. Chen [5], who showed that, for all $k\geq 2$,

[TABLE]

The holonomy Chen ranks of the group $G$ are defined as $\bar{\theta}_{k}(G):=\dim(\mathfrak{h}/\mathfrak{h}^{\prime\prime})_{k}$, where $\mathfrak{h}=\mathfrak{h}(G)$. It is readily seen that $\bar{\theta}_{k}(G)\geq\theta_{k}(G)$, with equality for $k\leq 2$. A basic result in the subject reads as follows: If $G$ is $1$-formal, then

[TABLE]

for all $k\geq 1$. This result was proved in [31, Cor. 9.4] for $1$-formal groups admitting a finite, commutator-relators presentation, and in full generality in [43, Prop. 8.1 and Cor. 8.6].

Proposition 8.6**.**

Let $G=F/\langle r\rangle$ be a one-relator group, where $F=\langle x_{1},\dots,x_{n}\rangle$, and suppose $G$ is $1$-formal. Then

[TABLE]

Proof.

First assume that $r\in[F,F]$. The claim is then proved in [31, Thm. 7.3].

Now assume that $r\notin[F,F]$. In that case, Theorem 5.5 implies that $\mathfrak{h}(G)\cong\mathfrak{lie}(y_{1},\dots,y_{n-1})$, which in turn is isomorphic to $\mathfrak{h}(F_{n-1})$. Since both $G$ are $F_{n-1}$ is $1$-formal, formula (41) implies that

[TABLE]

The claim then follows from Chen’s formula (40). ∎

8.3. Surface groups

The Riemann surface $\Sigma_{g}$ is a compact Kähler manifold, and thus, a formal space. The formality of $\Sigma_{g}$ also implies the $1$-formality of $\Pi_{g}$. As a consequence, the associated graded Lie algebra $\operatorname{gr}(\Pi_{g})$ is isomorphic to the holonomy Lie algebra $\mathfrak{h}(\Pi_{g})$, which has a presentation \mathfrak{h}(\Pi_{g})=\mathfrak{lie}(2g)\Big{/}\Big{\langle}\sum_{i=1}^{g}[x_{i},y_{i}]=0\Big{\rangle}, where $\mathfrak{lie}(2g):=\mathfrak{lie}(x_{1},y_{1},\dots,x_{g},y_{g})$. It follows from formula (36) that the LCS ranks of the $1$-relator group $\Pi_{g}$ are given by

[TABLE]

Using now Theorem 5.11, we see that the Chen Lie algebra of $\Pi_{g}$ has presentation

[TABLE]

Furthermore, Proposition 8.6 shows that the Chen ranks of our surface group are given by $\theta_{1}(\Pi_{g})=2g$, $\theta_{2}(\Pi_{g})=2g^{2}-g-1$, and

[TABLE]

Let $N_{h}$ be the nonorientable surface of genus $h\geq 1$. It is well known that $N_{h}$ has the rational homotopy type of a wedge of $h-1$ circles, see [7, Example 6.18]. Hence, $N_{h}$ is a formal space, and thus $\pi_{1}(N_{h})$ is a $1$-formal group. Furthermore, Proposition 8.1 shows that the holonomy Lie algebra of $\pi_{1}(N_{h})$ is isomorphic to the free Lie algebra with $h-1$ generators, and Proposition 8.6 implies that the Chen ranks of $\pi_{1}(N_{h})$ are given by $\theta_{k}(\pi_{1}(N_{h}))=(k-1)\binom{h+k-3}{k}$ for $k\geq 2$.

9. Seifert fibered spaces

We will consider here only orientable, closed Seifert manifolds with orientable base. Every such manifold $M$ admits an effective circle action, with orbit space an orientable surface of genus $g$, and finitely many exceptional orbits, encoded in pairs of coprime integers $(\alpha_{1},\beta_{1}),\dots,(\alpha_{s},\beta_{s})$ with $\alpha_{j}\geq 2$. The obstruction to trivializing the bundle $\eta\colon M\to\Sigma_{g}$ outside tubular neighborhoods of the exceptional orbits is given by an integer $b=b(\eta)$. A standard presentation for the fundamental group of $M$ in terms of the Seifert invariants is given by

[TABLE]

As shown by Scott in [39], the Euler number $e(\eta)$ of the Seifert bundle $\eta\colon M\rightarrow\Sigma_{g}$ satisfies $e(\eta)=-b(\eta)-\sum_{i=1}^{s}\beta_{i}/\alpha_{i}$.

9.1. Holonomy Lie algebra

We now give a presentation for the holonomy Lie algebra of a Seifert manifold group.

Theorem 9.1**.**

Let $\eta\colon M\to\Sigma_{g}$ be a Seifert fibration with orientable base. The rational holonomy Lie algebra of the group $\pi_{\eta}=\pi_{1}(M)$ is given by

[TABLE]

Proof.

First assume $e(\eta)=0$. In this case, the row-echelon approximation of $\pi_{\eta}$ has presentation

[TABLE]

It is readily seen that the rank of the Jacobian matrix associated to this presentation has rank $s$. Furthermore, the map $\pi\colon F_{\mathbb{Q}}\rightarrow H_{\mathbb{Q}}$ is given by $x_{i}\mapsto x_{i}$, $y_{i}\mapsto y_{i}$, $z_{j}\mapsto(-\beta_{i}/\alpha_{i})h$, $h\mapsto h$. Let $\kappa$ be the Magnus expansion from Definition 2.1. A Fox Calculus computation shows that $\kappa$ takes the following values on the commutator-relators of $\widetilde{\pi}_{\eta}$: $\kappa([z_{i},h])=1,$

[TABLE]

where $r=([x_{1},y_{1}]\cdots[x_{g},y_{g}])^{\alpha_{1}\cdots\alpha_{s}}$. The first claim now follows from Theorem 5.5.

Next, assume $e(\eta)\neq 0$. Then the row-echelon approximation of $\pi_{\eta}$ is given by

[TABLE]

while the homomorphism $\pi\colon F_{\mathbb{Q}}\rightarrow H_{\mathbb{Q}}$ is given by $x_{i}\mapsto x_{i}$, $y_{i}\mapsto y_{i}$, $z_{j}\mapsto(-\beta_{i}/\alpha_{i})h$, and $h\mapsto 0$. As before, the second claim follows from Theorem 5.5, and we are done. ∎

The Malcev Lie algebra of $\pi_{\eta}$, given in [41, Thm.11.6], has an explicit presentation, which is the degree completion of the graded Lie algebra

[TABLE]

where $\deg(w)=2$ and the other generators have degree $1$. Moreover, $\operatorname{gr}(\pi_{\eta})\cong L(\pi_{\eta})$. From the presentation of $\pi_{\eta}$ and the definition of filtered formality, we immediately obtain that fundamental groups of orientable Seifert manifolds are filtered-formal.

9.2. LCS ranks

We end this section with a computation of the ranks of the various graded Lie algebras attached to the fundamental group of an orientable Seifert manifold. Comparing these ranks, we derive some consequences regarding the non-formality properties of such groups.

We start with the LCS ranks $\phi_{k}(\pi_{\eta})=\dim\operatorname{gr}_{k}(\pi_{\eta})$ and the holonomy ranks $\bar{\phi}_{k}(\pi_{\eta})=\dim\mathfrak{h}(\pi_{\eta})_{k}$.

Proposition 9.2**.**

The LCS ranks and the holonomy ranks of a Seifert manifold group $\pi_{\eta}$ are computed as follows.

1. (1)

If $e(\eta)=0$, then $\phi_{1}(\pi_{\eta})=\bar{\phi}_{1}(\pi_{\eta})=2g+1$, and $\phi_{k}(\pi_{\eta})=\bar{\phi}_{k}(\pi_{\eta})=\phi_{k}(\Pi_{g})$ for $k\geq 2$. 2. (2)

If $e(\eta)\neq 0$, then $\bar{\phi}_{k}(\pi_{\eta})=\phi_{k}(F_{2g})$ for $k\geq 1$. 3. (3)

If $e(\eta)\neq 0$, then $\phi_{1}(\pi_{\eta})=2g$, $\phi_{2}(\pi_{\eta})=g(2g-1)$, and $\phi_{k}(\pi_{\eta})=\phi_{k}(\Pi_{g})$ for $k\geq 3$.

Here the LCS ranks $\phi_{k}(\Pi_{g})$ are given by formula (43).

Proof.

If $e(\eta)=0$, then $\pi_{\eta}\cong\Pi_{g}\times\mathbb{Z}$, and claim (1) readily follows. So suppose that $e(\eta)\neq 0$. In this case, we know from Theorem 9.1 that $\mathfrak{h}(\pi_{\eta})=\mathfrak{h}(F_{2g})$, and thus claim (2) follows.

By (49), the associated graded Lie algebra $\operatorname{gr}(\pi_{\eta})$ is isomorphic to the quotient of the free Lie algebra $\mathfrak{lie}(x_{1},y_{1},\dots,x_{g},y_{g},w)$ by the ideal generated by the elements $\sum_{i=1}^{g}[x_{i},y_{i}]-w$, $[w,x_{i}]$, and $[w,y_{i}]$. Define a morphism $\chi\colon\operatorname{gr}(\pi_{\eta})\to\operatorname{gr}(\Pi_{g})$ by sending $x_{i}\mapsto x_{i}$, $y_{i}\mapsto y_{i}$, and $w\mapsto 0$. It is readily seen that the kernel of $\chi$ is the Lie ideal of $\operatorname{gr}(\pi_{\eta})$ generated by $w$, and this ideal is isomorphic to the free Lie algebra on $w$. Thus, we get a short exact sequence of graded Lie algebras,

[TABLE]

Comparing Hilbert series in this sequence establishes claim (3) and completes the proof. ∎

Corollary 9.3**.**

If $g=0$, the group $\pi_{\eta}$ is always $1$-formal, while if $g>0$, the group $\pi_{\eta}$ is graded-formal if and only if $e(\eta)=0$.

Proof.

First suppose $e(\eta)=0$. In this case, we know from (49) that $\operatorname{gr}(\pi_{\eta})\cong\operatorname{gr}(\Pi_{g})\times\operatorname{gr}(\mathbb{Z})$. It easily follows that $\operatorname{gr}(\pi_{\eta})\cong\mathfrak{h}(\pi_{\eta})$ by comparing the presentations of these two Lie algebras. Hence, $\pi_{\eta}$ is graded-formal, and thus $1$-formal, by the fact that $\pi_{\eta}$ is filtered formal.

It is enough to assume that $g>0$ and $e(\eta)\neq 0$, since the other claims are clear. By Proposition 9.2, we have that $\bar{\phi}_{3}(\pi_{\eta})=(8g^{3}-2g)/3$, whereas $\phi_{3}(\pi_{\eta})=(8g^{3}-8g)/3$. Hence, $\mathfrak{h}(\pi_{\eta})$ is not isomorphic to $\operatorname{gr}(\pi_{\eta})$, proving that $\pi_{\eta}$ is not graded-formal. ∎

9.3. Chen ranks

Recall that the Chen ranks are defined as $\theta_{k}(\pi)=\dim\operatorname{gr}_{k}(\pi/\pi^{\prime\prime})$, while the holonomy Chen ranks are defined as $\bar{\theta}_{k}(\pi)=\dim(\mathfrak{h}/\mathfrak{h}^{\prime\prime})_{k}$, where $\mathfrak{h}=\mathfrak{h}(\pi)$.

Proposition 9.4**.**

The Chen ranks and the holonomy Chen ranks of a Seifert manifold group $\pi_{\eta}$ are computed as follows.

1. (1)

If $e(\eta)=0$, then $\theta_{1}(\pi_{\eta})=\bar{\theta}_{1}(\pi_{\eta})=2g+1$, and $\theta_{k}(\pi_{\eta})=\bar{\theta}_{k}(\pi_{\eta})=\theta_{k}(\Pi_{g})$ for $k\geq 2$. 2. (2)

If $e(\eta)\neq 0$, then $\bar{\theta}_{k}(\pi_{\eta})=\theta_{k}(F_{2g})$ for $k\geq 1$. 3. (3)

If $e(\eta)\neq 0$, then $\theta_{1}(\pi_{\eta})=2g$, $\theta_{2}(\pi_{\eta})=g(2g-1)$, and $\theta_{k}(\pi_{\eta})=\theta_{k}(\Pi_{g})$ for $k\geq 3$.

Here the Chen ranks $\theta_{k}(F_{2g})$ and $\theta_{k}(\Pi_{g})$ are given by formulas (40) and (45), respectively.

Proof.

Claims (1) and (2) are easily proved, as in Proposition 9.2. To prove claim (3), start by recalling that the group $\pi_{\eta}$ is filtered-formal. Hence, from [41, Theorem 9.3], the Chen Lie algebra $\operatorname{gr}(\pi_{\eta}/\pi_{\eta}^{\prime\prime})$ is isomorphic to $\operatorname{gr}(\pi_{\eta})/\operatorname{gr}(\pi_{\eta})^{\prime\prime}$. As before, we obtain a short exact sequence of graded Lie algebras,

[TABLE]

Comparing Hilbert series in this sequence completes the proof. ∎

Remark 9.5**.**

The above result can be used to give another proof of Corollary 9.3. Indeed, suppose $e(\eta)\neq 0$. Then, by Proposition 9.4, we have that $\bar{\theta}_{3}(\pi_{\eta})-\theta_{3}(\pi_{\eta})=2g$. Consequently, the group $\pi_{\eta}$ is not $1$-formal. The group $\pi_{\eta}$ is not graded-formal, since it is filtered-formal.