The Lie algebra associated with the lower central series of a right-angled Coxeter group

TL;DR

This paper explores the structure of the Lie algebra associated with the lower central series of right-angled Coxeter groups, providing explicit combinatorial descriptions for initial factors.

Contribution

It offers a new explicit combinatorial description of the first three factors of the lower central series for right-angled Coxeter groups.

Findings

Explicit description of the first three factors of the lower central series.

Connection established between the Lie algebra of the group and the graph Lie algebra.

Enhanced understanding of the algebraic structure of right-angled Coxeter groups.

Abstract

We study the lower central series of a right-angled Coxeter group $RC_K$ and the associated Lie algebra $L(RC_K)$. The latter is related to the graph Lie algebra $L_K$. We give an explicit combinatorial description of the first three consecutive factors of the lower central series of the group $RC_K$.

Equations

$$\mbox{\it RC\/}_{\mathcal{K}}=F(g_{1},\ldots,g_{m})\big{/}(g_{i}^{2}=1\text{ for }i\in\{1,\ldots,m\},\;\;g_{i}g_{j}=g_{j}g_{i}\text{ when }\{i,j\}\in\mathcal{K}),$$

$$\mbox{\it RA\/}_{\mathcal{K}}=F(g_{1},\ldots,g_{m})\big{/}(g_{i}g_{j}=g_{j}g_{i}\text{ when }\{i,j\}\in\mathcal{K}).$$

$$(\text{X},\text{A})=\{(X_1,A_1),\dots ,(X_m,A_m)\}$$

$$({\textbf{{X}}},{\textbf{{A}}})^{I}=\bigl{\{}(x_{1},\ldots,x_{m})\in\prod_{k=1}^{m}X_{k}\colon\;x_{k}\in A_{k}\quad\text{for }k\notin I\bigl{\}}$$

$$({\textbf{{X}}},{\textbf{{A}}})^{\mathcal{K}}=\bigcup_{I\in\mathcal{K}}({\textbf{{X}}},{\textbf{{A}}})^{I}=\bigcup_{I\in\mathcal{K}}\Bigl{(}\prod_{i\in I}X_{i}\times\prod_{i\notin I}A_{i}\Bigl{)},$$

$${\mathcal{R}}_{\mathcal{K}}=(D^1,S^0{)}^{\mathcal{K}}=\bigcup _{I\in \mathcal{K}}(D^1,S^0{)}^I.$$

$${\mathcal{K}}_J=\{I\in \mathcal{K}:I\subset J\},$$

$$H_k({\mathcal{R}}_{\mathcal{K}};\mathbb{Z})\cong \underset{J\subset [m]}{⨁}{\tilde{H}}_{k-1}({\mathcal{K}}_J)$$

$$(q_{i_1},q_{i_2},\dots ,q_{i_k}):=(\dots ((q_{i_1},q_{i_2}),q_{i_3}),\dots ,q_{i_k}).$$

$$[{\mu }_{i_1},{\mu }_{i_2},\dots ,{\mu }_{i_k}]:=[\dots [[{\mu }_{i_1},{\mu }_{i_2}],{\mu }_{i_3}],\dots ,{\mu }_{i_k}].$$

$$(a,bc)=(a,c)(a,b)(a,b,c),$$

$$(ab,c)=(a,c)(a,c,b)(b,c),$$

$$(a,b,c)(b,c,a)(c,a,b)=(b,a)(c,a)(c,b{)}^a(a,b)(a,c{)}^b(b,c{)}^a(a,c)(c,a{)}^b,$$

$$L(G):=\underset{k=1}{\overset{+\infty }{⨁}}{L}^k(G).$$

$$(g_i,g_j),(g_i,g_j,g_{k_1}),\dots ,(g_i,g_j,g_{k_1},g_{k_2},\dots ,g_{k_{m-2}}),$$

$$(r,l)=(r,l_2\cdot g_i^{\prime })=(r,g_i^{\prime })\cdot (r,l_2)\cdot ((r,l_2),g_i^{\prime }).$$

$$a_1{a}_2\cdots a_n{a}_1{a}_2\cdots a_n=(a_1^{-1},(a_2\cdots a_n{)}^{-1})\cdot (a_2\cdots a_n)a_1^2(a_2\cdots a_n{)}^{-1}\cdot (a_2\cdots a_n{)}^2.$$

$$L_{\mathcal{K}}:=\text{mbox}{FL\text{/}}_{{\mathbb{Z}}_2}⟨{\mu }_1,{\mu }_2,\dots ,{\mu }_n⟩/([{\mu }_i,{\mu }_j]=0\text{ for }\{i,j\}\in \mathcal{K}).$$

$$\tilde{\phi }:\text{mbox}{FL\text{/}}_{{\mathbb{Z}}_2}⟨{\mu }_1,{\mu }_2,\dots ,{\mu }_n⟩\to L(\text{mbox}{RC\text{/}}_{\mathcal{K}}),{\mu }_i\mapsto {\overline{g}}_i.$$

$$L^3(\text{mbox}{RC\text{/}}_{\mathcal{K}})\cong L_{\mathcal{K}}^3/([{\mu }_1,{\mu }_2,{\mu }_1]=[{\mu }_1,{\mu }_2,{\mu }_2]).$$

$$L(\mbox{\it RC\/}_{\mathcal{K}})\cong L_{\mathcal{K}}\big{/}\bigl{(}[a,\mu_{1}]=[a,\mu_{2}],\;\;[a,\underbrace{\mu_{1},\ldots,\mu_{1}}_{2k+1},a]=0,\;k\geqslant 0\bigr{)},$$

$$[\underset{n}{\underbrace{1212\dots 1212}}]=[[[\underset{n-2}{\underbrace{1212\dots 12}}],1],2]=[[2,[\underset{n-2}{\underbrace{1212\dots 12}}]],1]+[[2,1],[\underset{n-2}{\underbrace{1212\dots 12}}]]=$$

$$=[\underset{n-1}{\underbrace{1212\dots 122}}1]+[[21],[\underset{n-2}{\underbrace{1212\dots 12}}]]=[\underset{n}{\underbrace{1212\dots 1211}}]+[[21],[\underset{n-2}{\underbrace{1212\dots 12}}]],$$

$$[\underset{n}{\underbrace{1212\dots 121}}]=[[[\underset{n-2}{\underbrace{1212\dots 1}}],2],1]=[[1,[\underset{n-2}{\underbrace{1212\dots 1}}]],2]+[[2,1],[\underset{n-2}{\underbrace{1212\dots 1}}]]=$$

$$=[\underset{n-1}{\underbrace{1212\dots 11}}2]+[[21],[\underset{n-2}{\underbrace{1212\dots 1}}]]=[\underset{n}{\underbrace{1212\dots 122}}]+[[21],[\underset{n-2}{\underbrace{1212\dots 1}}]],$$

$$A_{n,i}:=[[a\underset{i+1}{\underbrace{bb\dots b}}],[a\underset{n-6-i}{\underbrace{bb\dots b}}]]\in L^{n-1}(\text{mbox}{RC\text{/}}_{\mathcal{K}}),-1⩽i⩽n-6.$$

$$0=A_{k,-1}=[a,[a\underset{k-5}{\underbrace{bb\dots b}}]]=[[a\underset{k-6}{\underbrace{bb\dots b}}a],b]+[[a,b],[a\underset{k-6}{\underbrace{bb\dots b}}]]=$$

$$=[A_{k-1,-1},b]+[[a,b],[a\underset{k-6}{\underbrace{bb\dots b}}]]=[[a,b],[a\underset{k-6}{\underbrace{bb\dots b}}]]=A_{k,0}.$$

$$A_{k,0}=[A_{k-1,0},b]+[[abb],[a\underset{k-7}{\underbrace{bb\dots b}}]]=A_{k,1},$$

$$A_{n+1,-1}=[a,[a\underset{n-4}{\underbrace{bb\dots b}}]]=[a\underset{n-4}{\underbrace{bb\dots b}}a]=[[[a\underset{n-5}{\underbrace{bb\dots b}}],b],a]=$$

$$=[[a\underset{n-5}{\underbrace{bb\dots b}}a],b]+[[a,b],[a\underset{n-5}{\underbrace{bb\dots b}}]]=[[\stackrel{\text{ = 0}}{\overbrace{A_{n,-1}}},b]+[[a,b],[a\underset{n-5}{\underbrace{bb\dots b}}]]=$$

Full text

The research is supported by the Russian Foundation for Basic Research (grants no. 16-51-55017, 17-01-00671).

1. Introduction

A right-angled Coxeter group $\mbox{\it RC\/}_{\mathcal{K}}$ is a group with $m$ generators $g_{1},\ldots,g_{m}$, satisfying the relations $g_{i}^{2}=1$ for all $i\in\{1,\ldots,m\}$ and the commutator relations $g_{i}g_{j}=g_{j}g_{i}$ for some pairs $\{i,j\}$. Such a group is determined by a graph $\mathcal{K}^{1}$ with $m$ vertices, where two vertices are connected by an edge if the corresponding generators commute. Right-angled Coxeter groups are classical objects in geometric group theory. In this paper we study the lower central series of a right-angled Coxeter group $\mbox{\it RC\/}_{\mathcal{K}}$ and the associated graded Lie algebra $L(\mbox{\it RC\/}_{\mathcal{K}})$.

Right-angled Artin groups $\mbox{\it RA\/}_{\mathcal{K}}$ differ from right-angled Coxeter groups $\mbox{\it RC\/}_{\mathcal{K}}$ by the absence of relations $g_{i}^{2}=1$. The associated Lie algebra $L(\mbox{\it RA\/}_{\mathcal{K}})$ was fully described in [4], see also [10], [7]. Namely, it was proved that the Lie algebra $L(\mbox{\it RA\/}_{\mathcal{K}})$ is a isomorphic to the graph Lie algebra (over $\mathbb{Z}$) corresponding to the graph $\mathcal{K}^{1}$.

For right-angled Coxeter groups, the quotient groups $\gamma_{1}(\mbox{\it RC\/}_{\mathcal{K}})/\gamma_{n}(\mbox{\it RC\/}_{\mathcal{K}})$ were described for some particular $\mathcal{K}$ and $n$ in [8], [9]. For $n\geqslant 4$ difficulties arose, similar to those we encountered in the calculation of the successive quotients $\gamma_{n-1}(\mbox{\it RC\/}_{\mathcal{K}})/\gamma_{n}(\mbox{\it RC\/}_{\mathcal{K}})$. In contrast to the case of right-angled Artin groups, the problem of describing the associated Lie algebra $L(\mbox{\it RC\/}_{\mathcal{K}})$ is much harder due to the lack of an isomorphism between the algebra $L(\mbox{\it RC\/}_{\mathcal{K}})$ and the graph Lie algebra $L_{\mathcal{K}}$ over $\mathbb{Z}_{2}$ (see Section 4). In this paper we construct an epimorphism of Lie algebras $L_{\mathcal{K}}\rightarrow L(\mbox{\it RC\/}_{\mathcal{K}})$ and in some cases describe its kernel (see Propositions 4.1–4.4). For an arbitrary group $\mbox{\it RC\/}_{\mathcal{K}}$, we give a combinatorial description of the bases for the first three graded components of the associated Lie algebra $L(\mbox{\it RC\/}_{\mathcal{K}})$ (see Theorem 4.5).

I express my gratitude to my supervisor Taras Evgenievich Panov for suggesting the problem, help and advice.

2. Preliminaries

Let $\mathcal{K}$ be an (abstract) simplicial complex on the set $[m]=\{1,2,\dots,m\}$. A subset $I=\{i_{1},\ldots,i_{k}\}\in\mathcal{K}$ is called a simplex (or face) of $\mathcal{K}$. We always assume that $\mathcal{K}$ contains $\varnothing$ and all singletons $\{i\}$, $i=1,\ldots,m$.

We denote by $F_{m}$ or $F(g_{1},\ldots,g_{m})$ a free group of rank $m$ with generators $g_{1},\ldots,g_{m}$.

The right-angled Coxeter (Artin) group $\mbox{\it RC\/}_{\mathcal{K}}$ ($\mbox{\it RA\/}_{\mathcal{K}}$) corresponding to $\mathcal{K}$ is defined by generators and relations as follows:

[TABLE]

[TABLE]

Clearly, the group $\mbox{\it RC\/}_{\mathcal{K}}$ ($\mbox{\it RA\/}_{\mathcal{K}}$) depends only on the $1$-skeleton of $\mathcal{K}$, the graph $\mathcal{K}^{1}$.

We recall the construction of polyhedral products.

Let $\mathcal{K}$ be a simplicial complex on $[m]$, and let

[TABLE]

be a sequence of $m$ pairs of pointed topological spaces, $\mathit{pt}\in A_{i}\subset X_{i}$, where $\mathit{pt}$ denotes the basepoint. For each subset $I\subset[m]$ we set

[TABLE]

and define the polyhedral product $({\textbf{{X}}},{\textbf{{A}}})^{\mathcal{K}}$ as

[TABLE]

where the union is taken inside the Cartesian product $\prod_{k=1}^{m}X_{k}$.

In the case when all pairs $(X_{i},A_{i})$ are the same, i. e. $X_{i}=X$ and $A_{i}=A$ for $i=1,\ldots,m$, we use the notation $(X,A)^{\mathcal{K}}$ for $({\textbf{{X}}},{\textbf{{A}}})^{\mathcal{K}}$. Also, if each $A_{i}=\mathit{pt}$, then we use the abbreviated notation ${\textbf{{X}}}^{\mathcal{K}}$ for $({\textbf{{X}}},\mathit{pt})^{\mathcal{K}}$, and $X^{\mathcal{K}}$ for $(X,\mathit{pt})^{\mathcal{K}}$.

For details on this construction and examples see [2, §3.5], [1], [3, §4.3].

Let $(X_{i},A_{i})=(D^{1},S^{0})$ for $i=1,\ldots,m$, where $D^{1}$ is a segment and $S^{0}$ is its boundary consisting of two points. The corresponding polyhedral product is known as the real moment-angle complex [2, §3.5], [3] and is denoted by $\mathcal{R}_{\mathcal{K}}$:

[TABLE]

We shall also need the polyhedral product $(\mathbb{R}P^{\infty})^{\mathcal{K}}$, where $\mathbb{R}P^{\infty}$ the infinite-dimensional real projective space.

A simplicial complex $\mathcal{K}$ is called a flag complex if any set of vertices of $\mathcal{K}$ which are pairwise connected by edges spans a simplex. Any flag complex $\mathcal{K}$ is determined by its one-dimensional skeleton $\mathcal{K}^{1}$.

The relationship between polyhedral products and right-angled Coxeter groups is described by the following result.

Theorem 2.1** (see [6, Corollary 3.4]).**

Let $\mathcal{K}$ be a simplicial complex on $m$ vertices.

• (a)

$\pi_{1}((\mathbb{R}P^{\infty})^{\mathcal{K}})\cong\mbox{\it RC\/}_{\mathcal{K}}$.

• (b)

Each of the spaces $(\mathbb{R}P^{\infty})^{\mathcal{K}}$ и $\mathcal{R}_{\mathcal{K}}$ is aspherical if and only if $\mathcal{K}$ is a flag complex.

• (c)

$\pi_{i}((\mathbb{R}P^{\infty})^{\mathcal{K}})\cong\pi_{i}(\mathcal{R}_{\mathcal{K}})$* for $i\geqslant 2$.*

• (d)

The group $\pi_{1}(\mathcal{R}_{\mathcal{K}})$ isomorphic to the commutator subgroup $\mbox{\it RC\/}^{\prime}_{\mathcal{K}}$.

For each subset $J\subset[m]$, consider the restriction of $\mathcal{K}$ to $J$:

[TABLE]

which is also known as a full subcomplex of $\mathcal{K}$.

The following theorem gives a combinatorial description of homology of the real moment-angle complex $\mathcal{R}_{\mathcal{K}}$:

Theorem 2.2** ([2], [3, §4.5]).**

There is an isomorphism

[TABLE]

for any $k\geqslant 0$, where $\widetilde{H}_{k-1}(\mathcal{K}_{J})$ is the reduced simplicial homology group of  $\mathcal{K}_{J}$.

If $\mathcal{K}$ is a flag complex, then Theorem 2.2 also gives a description of the integer homology groups of the commutator subgroup $\mbox{\it RC\/}_{\mathcal{K}}^{\prime}$.

Let $G$ be group. The commutator of two elements $a,b\in G$ given by the formula $(a,b)=a^{-1}b^{-1}ab$.

We refer to the following nested commutator of length $k$

[TABLE]

as the simple nested commutator of $q_{i_{1}},q_{i_{2}},\ldots,q_{i_{k}}$.

Similarly, we define simple nested Lie commutators

[TABLE]

For any group $G$ and any three elements $a,b,c\in G$, the following Hall–Witt identities hold:

[TABLE]

where $a^{b}=b^{-1}ab$.

Let $H,W\subset G$ be subgroups. Then we define $(H,W)\subset G$ as the subgroup generated by all commutators $(h,w),h\in H,w\in W$. In particular, the commutator subgroup $G^{\prime}$ of the group $G$ is $(G,G)$.

For any group $G$, set $\gamma_{1}(G)=G$ and define inductively $\gamma_{k+1}(G)=(\gamma_{k}(G),G)$. The resulting sequence of groups $\gamma_{1}(G),\gamma_{2}(G),\ldots,\gamma_{k}(G),\ldots$ is called the lower central series of $G$.

If $H\subset G$ is normal subgroup, i. e. $H=g^{-1}Hg$ for all $g\in G$, we will use the notation $H\lhd G$.

In particular, $\gamma_{k+1}(G)\lhd\gamma_{k}(G)$, and the quotient group $\gamma_{k}(G)/\gamma_{k+1}(G)$ is abelian. Denote $L^{k}(G):=\gamma_{k}(G)/\gamma_{k+1}(G)$ and consider the direct sum

[TABLE]

Given an element $a_{k}\in\gamma_{k}(G)\subset G$, we denote by $\overline{a}_{k}$ its conjugacy class in the quotient group $L^{k}(G)$. If $a_{k}\in\gamma_{k}(G),\;a_{l}\in\gamma_{l}(G)$, then $(a_{k},a_{l})\in\gamma_{k+l}(G)$. Then the Hall–Witt identities imply that $L(G)$ is a graded Lie algebra over $\mathbb{Z}$ (a Lie ring) with Lie bracket $[\overline{a}_{k},\overline{a}_{l}]:=\overline{(a_{k},a_{l})}$. The Lie algebra $L(G)$ is called the Lie algebra associated with the lower central series (or the associated Lie algebra) of $G$.

Theorem 2.3** ([6, Theorem 4.5]).**

Let $\mbox{\it RC\/}_{\mathcal{K}}$ be right-angled Coxeter group corresponding to a simplicial complex $\mathcal{K}$ with $m$ vertices. Then the commutator subgroup $\mbox{\it RC\/}^{\prime}_{\mathcal{K}}$ has a finite minimal set of generators consisting of $\sum_{J\subset[m]}\mathop{\mathrm{rank}}\widetilde{H}_{0}(\mathcal{K}_{J})$ nested commutators

[TABLE]

where $i<j>k_{1}>k_{2}>\ldots>k_{\ell-2}$, $k_{s}\neq i$ for all $s$, and $i$ is the smallest vertex in a connected component not containing $j$ of the subcomplex $\mathcal{K}_{\{k_{1},\ldots,k_{\ell-2},j,i\}}$.

Remark*.*

In [6] commutators were nested to the right. Now we nest them to the left.

From Theorems 2.2 and 2.3 we get:

Corollary 2.4**.**

The group $H_{1}(\mathcal{R}_{\mathcal{K}})=\mbox{\it RC\/}_{\mathcal{K}}^{\prime}/\mbox{\it RC\/}_{\mathcal{K}}^{\prime\prime}$ is a free abelian group of rank $\sum_{J\subset[m]}\mathop{\mathrm{rank}}\widetilde{H}_{0}(\mathcal{K}_{J})$ with basis consisting of the images of the iterated commutators described in Theorem 2.3.

3. The lower central series of a right-angled Coxeter group

Let $G$ be a group with generators $g_{i},i\in I$. An element $a\in G$ can be written (generally, not uniquely) as a word $w=(g_{i_{1}})^{t_{1}}(g_{i_{2}})^{t_{2}}\cdots(g_{i_{p}})^{t_{p}}$, where $t_{i}\in\mathbb{Z}$. The length of $w$ is defined as $len(w)=\sum_{i=1}^{p}|t_{i}|$.

There is the following standard result (see [5, § 5.3]):

Proposition 3.1**.**

Let $G$ be a group with generators $g_{i},i\in I$. The $k$-th term $\gamma_{k}(G)$ of the lower central series is generated by simple nested commutators of length greater than or equal to $k$ in generators and their inverses.

Proof.

Apply induction on $k$. The base $k=1$ is obvious.

Assuming the statement holds for some $k$, we prove it for $k+1$. Each element of $\gamma_{k+1}(G)$ is represented as a product of commutators of the form $(r,l)$ and $(l,r)$, where $r\in\gamma_{k}(G),l\in G$. Therefore, it suffices to prove that any commutator $(r,l)$ and any commutator $(l,r)$, $r\in\gamma_{k}(G)$, $l\in G$, can be written in the required form. We write $g_{i}^{\prime}$ for either $g_{i}$ or $g_{i}^{-1}$. Write the element $l$ as a word $l_{1}$ in the generators $g_{i}$, and write $l_{1}=l_{2}\cdot g_{i}^{\prime}$ for some $i$, so that $len(l_{2})<len(l_{1})$. Apply identities (2.3):

[TABLE]

By writing the element $r\in\gamma_{k}(G)$ as a product of commutators of length $\geqslant k$ and consequently applying identities (2.3) to $(r,g_{i}^{\prime})$, we obtain a product of commutators of length $\geqslant k+1$. Then we apply the same procedure to $(r,l_{2})$. Since $len(l_{2})<len(l_{1})=len(l)$ and $len(l)<+\infty$, continuing this process, we eventually come to a product of commutators of length $\geqslant k+1$; this follows from the fact that the length of the commutators does not decrease after applying identities (2.3). For the bracket $((r,l_{2}),g_{i}^{\prime})$, we use identities (2.3) to expand the commutator $(A,g_{i}^{\prime})$, where $A$ is an expression of $(r,l_{2})$ via commutators of length $\geqslant k$. Since the application of identities (2.3) does not decrease the length of the commutators, we obtain a product of commutators of length $\geqslant k+1$. The argument for $(l,r)$ is similar to $(r,l)$. ∎

Corollary 3.2**.**

Let $\mbox{\it RC\/}_{\mathcal{K}}$ be a right-angled Coxeter group with generators $g_{i}$. Then the group $\gamma_{k}(\mbox{\it RC\/}_{\mathcal{K}})$ is generated by commutators of length greater than or equal to $k$ in generators $g_{i}$.

Proof.

In the case of a right-angled Coxeter group we have $g_{i}^{-1}=g_{i}$. ∎

Proposition 3.3**.**

The square of any element of $\gamma_{k}(\mbox{\it RC\/}_{\mathcal{K}})$ is contained in $\gamma_{k+1}(\mbox{\it RC\/}_{\mathcal{K}})$.

Proof.

We use $\gamma_{k}$ instead of $\gamma_{k}(\mbox{\it RC\/}_{\mathcal{K}})$ in this proof.

Let $a\in\gamma_{k}$. If $k=1$, then $a=\prod_{i=1}^{n}g_{k_{i}}$. If $k>1$, then $a=\prod_{i=1}^{n}a_{i}$, where $a_{i}=(b_{i},g_{p_{i}})$ or $a_{i}=(g_{p_{i}},b_{i})$, $b_{i}\in\gamma_{k-1}$. We use induction on $n$.

Let $n=1$. The case $k=1$ is obvious (because $g_{k}^{2}=1$). If $k>1$, then $a=(b,g_{i})$ or $a=(g_{i},b)$ for some $b\in\gamma_{k-1}$. For $a=(b,g_{i})$ we have $a^{2}=(b,g_{i})(b,g_{i})=(g_{i},(b,g_{i}))\in\gamma_{k+1}$, and for $a=(g_{i},b)$ we have $a^{2}=(g_{i},b)(g_{i},b)=(g_{i},(g_{i},b))\in\gamma_{k+1}$.

Suppose now the statement is proved for $n-1$. Let $a=\prod_{i=1}^{n}a_{i}$ and $a^{2}=\prod_{i=1}^{n}a_{i}\cdot\prod_{i=1}^{n}a_{i}$. We have:

[TABLE]

Clearly, the first factor lies in $\gamma_{2k}\subset\gamma_{k+1}$. The second factor lies in $\gamma_{k+1}$ as a conjugate to $a_{1}^{2}$ (by induction). The last factor also lies in $\gamma_{k+1}$ by induction. ∎

4. Graded components of the associated Lie algebra $L(\mbox{\it RC\/}_{\mathcal{K}})$

In this section we consider the associated Lie algebra $L(\mbox{\it RC\/}_{\mathcal{K}})$ of a right-angled Coxeter group.

Here is an immediate corollary of Proposition 3.3:

Proposition 4.1**.**

$L(\mbox{\it RC\/}_{\mathcal{K}})$* is a Lie algebra over $\mathbb{Z}_{2}$.*

Hereinafter $\mathbb{Z}_{2}$ is a field of two elements.

We denote by $\mbox{\it FL\/}_{\mathbb{Z}_{2}}\langle\mu_{1},\mu_{2},\ldots,\mu_{n}\rangle$ a free graded Lie algebra over $\mathbb{Z}_{2}$ with $n$ generators $\mu_{i}$, where $\deg\mu_{i}=1$.

For any simplicial complex $\mathcal{K}$ we consider the graph Lie algebra over $\mathbb{Z}_{2}$:

[TABLE]

Clearly, $L_{\mathcal{K}}$ depends only on the $1$-skeleton $\mathcal{K}^{1}$ (a graph), however, as in the case of right-angled Coxeter groups, it is more convenient for us to work with simplicial complexes.

Proposition 4.2**.**

There is an epimorphism of Lie algebras $\varphi:L_{\mathcal{K}}\rightarrow L(\mbox{\it RC\/}_{\mathcal{K}})$.

Proof.

According to Proposition 4.1, $L(\mbox{\it RC\/}_{\mathcal{K}})$ is a Lie algebra over $\mathbb{Z}_{2}$, generated by the elements $\overline{g}_{i}\in\gamma_{1}(\mbox{\it RC\/}_{\mathcal{K}})/\gamma_{2}(\mbox{\it RC\/}_{\mathcal{K}}),i=1,\ldots,m$. By definition of a free Lie algebra, we have an epimorphism

[TABLE]

Since there is a relation $[\overline{g}_{i},\overline{g}_{j}]=0$ for $\{i,j\}\in\mathcal{K}$ in the Lie algebra $L(\mbox{\it RC\/}_{\mathcal{K}}$), the epimorphism $\widetilde{\varphi}$ factors through a required epimorphism $\varphi$. ∎

In fact, the homomorphism $\varphi$ from the proposition above is not injective, and the Lie algebras $L_{\mathcal{K}}$ and $L(\mbox{\it RC\/}_{\mathcal{K}})$ are not isomorphic. This distinguishes the case of right-angled Coxeter groups from the case of the right-angled Artin groups, where the associated Lie algebra $L(\mbox{\it RA\/}_{\mathcal{K}})$ is isomorphic to the graph Lie algebra over $\mathbb{Z}$, see [4], [10], [7].

Example 4.3**.**

Let $\mathcal{K}$ consist of two disjoint points, i. e. $\mathcal{K}=\{1,2\}$. Then $L_{\mathcal{K}}=\mbox{\it FL\/}_{\mathbb{Z}_{2}}\langle\mu_{1},\mu_{2}\rangle=\mbox{\it FL\/}_{\mathbb{Z}_{2}}\langle\mu_{1}\rangle\ast\mbox{\it FL\/}_{\mathbb{Z}_{2}}\langle\mu_{2}\rangle$ (hereinafter $\ast$ denotes the free product of Lie algebras or groups). The lower central series of $\mbox{\it RC\/}_{\mathcal{K}}=\mathbb{Z}_{2}\ast\mathbb{Z}_{2}$ is as follows: $\gamma_{1}(\mbox{\it RC\/}_{\mathcal{K}})=\mathbb{Z}_{2}\ast\mathbb{Z}_{2}$, and for $k\geqslant 2$ we have $\gamma_{k}(\mbox{\it RC\/}_{\mathcal{K}})\cong\mathbb{Z}$ is an infinite cyclic group generated by the commutator $(g_{1},g_{2},g_{1},\ldots,g_{1})$ of length $k$. Proposition 3.3 implies that $\gamma_{k}(\mbox{\it RC\/}_{\mathcal{K}})/\gamma_{k+1}(\mbox{\it RC\/}_{\mathcal{K}})=\mathbb{Z}_{2}$ for $k>1$, and $\gamma_{1}(\mbox{\it RC\/}_{\mathcal{K}})/\gamma_{2}(\mbox{\it RC\/}_{\mathcal{K}})=\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}$. Consider the algebra $L(\mbox{\it RC\/}_{\mathcal{K}})$. From the arguments above, $L(\mbox{\it RC\/}_{\mathcal{K}})=(\mathbb{Z}_{2}\oplus\mathbb{Z}_{2})\oplus\mathbb{Z}_{2}\oplus\cdots\oplus\mathbb{Z}_{2}\oplus\cdots$. It is easy to see that $L^{k}(\mbox{\it RC\/}_{\mathcal{K}})\cong L^{k}_{\mathcal{K}}$ for $k=1,2$. However, $L^{3}_{\mathcal{K}}\cong\mathbb{Z}_{2}\langle[\mu_{1},\mu_{2},\mu_{1}],[\mu_{1},\mu_{2},\mu_{2}]\rangle$, while $L^{3}(\mbox{\it RC\/}_{\mathcal{K}})\cong\mathbb{Z}_{2}$. Therefore,

[TABLE]

It follows that the homomorphism $\varphi$ from Proposition 4.2 is not injective.

Proposition 4.4**.**

Let $\mathcal{K}$ consist of two disjoint points. Then

[TABLE]

where $a=[\mu_{1},\mu_{2}]$.

Proof.

In this proof, we denote a commutator of the form $[\mu_{p_{1}},\mu_{p_{2}},\ldots,\mu_{p_{l}}]$ by $[p_{1}p_{2}\ldots p_{l}]$. We use induction on the dimension $n$ of the graded components. The base $n=3,4,5,6,7$ is verified easily.

Assume the statement is proved for dimensions less than $n$. By induction, $L^{n}(\mbox{\it RC\/}_{\mathcal{K}})$ generated by the two elements $x:=[\underbrace{1212\ldots 1211}_{n}]$ and $y:=[\underbrace{1212\ldots 1212}_{n}]$ for even $n$, and by the two elements $x:=[\underbrace{1212\ldots 121}_{n}]$ and $y:=[\underbrace{1212\ldots 122}_{n}]$ for odd $n$. On the other hand, we have seen in Example 4.3 that $L^{n}(\mbox{\it RC\/}_{\mathcal{K}})\cong\mathbb{Z}_{2}$ when $n\geqslant 2$. Below we show that for even $n>3$ the relation $x=y$ follows from the relations $[a,\mu_{1}]=[a,\mu_{2}]$ and $[a,\underbrace{\mu_{1},\ldots,\mu_{1}}_{2k+1},a]=0$ where the commutator length is less than $n$ (i. e. $k<\frac{n-5}{2}$), and for odd $n>3$ we need to add a new relation $[a,\mu_{1},\ldots,\mu_{1},a]=0$ where is the commutator has length $n$.

For even $n$ we have

[TABLE]

[TABLE]

and for odd $n$ we have

[TABLE]

[TABLE]

where the last equality in both cases follows from the inductive hypothesis. We obtain that $x-y=[[21],[\underbrace{1212\ldots 12}_{n-2}]]$ for even $n$ and $x-y=[[21],[\underbrace{1212\ldots 121}_{n-2}]]$ for odd $n$. It follows from the induction hypothesis that $[\underbrace{1212\ldots 12}_{k}]=[\underbrace{1211\ldots 11}_{k}]$ for any even $k<n$, and $[\underbrace{1212\ldots 21}_{k}]=[\underbrace{1211\ldots 11}_{k}]$ for any odd $k<n$. We introduce the notation $a=[12],b=\mu_{1}$. In the new notation, $x-y=[a,[a\underbrace{bb\ldots b}_{n-4}]]$. Denote

[TABLE]

Then $x-y=A_{n+1,-1}$ and, by the induction hypothesis, $A_{k,-1}=0$ for all $k<n+1$. From the Jacobi identity and induction we have:

[TABLE]

[TABLE]

Hence, $A_{k,0}=0$ for $k<n+1$. Next, we have

[TABLE]

hence $A_{k,1}=0$. Continuing in this fashion, we obtain $A_{k,i}=0$ for $-1\leqslant i\leqslant k-6$ and $k<n+1$. Now we have

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Continuing in the same way, we find that $A_{n+1,i}=A_{n+1,j}$ for any $i,j$. If $n=2m$ ($n$ is even), then we have

[TABLE]

If $n$ is odd, then we need to add a new relation

[TABLE]

∎

The following theorem describes the first three consecutive quotients of the lower central series of a right-angled Coxeter group $\mbox{\it RC\/}_{\mathcal{K}}$.

Theorem 4.5**.**

Let $\mathcal{K}$ be a simplicial complex on $[m]$, let $\mbox{\it RC\/}_{\mathcal{K}}$ be the right-angled Coxeter group corresponding to $\mathcal{K}$, and $L(\mbox{\it RC\/}_{\mathcal{K}})$ its associated Lie algebra. Then:

• (a)

$L^{1}(\mbox{\it RC\/}_{\mathcal{K}})$* has a basis $\overline{g}_{1},\ldots,\overline{g}_{m}$;*

• (b)

$L^{2}(\mbox{\it RC\/}_{\mathcal{K}})$* has a basis consisting of the commutators $[\overline{g}_{i},\overline{g}_{j}]$ with $i<j$ and ${\{i,j\}\notin\mathcal{K}}$;*

• (c)

$L^{3}(\mbox{\it RC\/}_{\mathcal{K}})$* has a basis consisting of*

• –

the commutators $[\overline{g}_{i},\overline{g}_{j},\overline{g}_{j}]$ with $i<j$ and $\{i,j\}\notin\mathcal{K}$;

• –

the commutators $[\overline{g}_{i},\overline{g}_{j},\overline{g}_{k}]$ where $i<j>k,i\neq k$ and $i$ is the smallest vertex in a connected component of $\mathcal{K}_{\{i,j,k\}}$ not containing $j$.

Example 4.6**.**

Consider simplicial complexes on $3$ vertices.

Let \mathcal{K}=\begin{picture}(10.0,5.0)\put(0.0,2.0){\circle*{1.0}} \put(5.0,2.0){\circle*{1.0}} \put(10.0,2.0){\circle*{1.0}} \put(-0.5,-1.0){\scriptsize 1} \put(4.5,-1.0){\scriptsize 2} \put(9.5,-1.0){\scriptsize 3} \end{picture}\,\,\,. Then $L^{3}(\mbox{\it RC\/}_{\mathcal{K}})$ has a basis consisting of $5$ commutators: $[\overline{g}_{1},\overline{g}_{2},\overline{g}_{2}],[\overline{g}_{2},\overline{g}_{3},\overline{g}_{3}],[\overline{g}_{1},\overline{g}_{3},\overline{g}_{3}],[\overline{g}_{1},\overline{g}_{3},\overline{g}_{2}],[\overline{g}_{2},\overline{g}_{3},\overline{g}_{1}]$.

Let \mathcal{K}=\begin{picture}(10.0,5.0)\put(0.0,2.0){\circle*{1.0}} \put(5.0,2.0){\circle*{1.0}} \put(10.0,2.0){\circle*{1.0}} \put(0.0,2.0){\line(1,0){5.0}} \put(-0.5,-1.0){\scriptsize 1} \put(4.5,-1.0){\scriptsize 2} \put(9.5,-1.0){\scriptsize 3} \end{picture}\,\,\,. Then $L^{3}(\mbox{\it RC\/}_{\mathcal{K}})$ has a basis consisting of $3$ commutators: $[\overline{g}_{2},\overline{g}_{3},\overline{g}_{3}],[\overline{g}_{1},\overline{g}_{3},\overline{g}_{3}],[\overline{g}_{1},\overline{g}_{3},\overline{g}_{2}]$.

Let \mathcal{K}=\begin{picture}(10.0,5.0)\put(0.0,2.0){\circle*{1.0}} \put(5.0,2.0){\circle*{1.0}} \put(10.0,2.0){\circle*{1.0}} \put(0.0,2.0){\line(1,0){5.0}} \put(5.0,2.0){\line(1,0){5.0}} \put(-0.5,-1.0){\scriptsize 1} \put(4.5,-1.0){\scriptsize 2} \put(9.5,-1.0){\scriptsize 3} \end{picture}\,\,\,. Then $L^{3}(\mbox{\it RC\/}_{\mathcal{K}})$ is generated by the commutator $[\overline{g}_{1},\overline{g}_{3},\overline{g}_{3}]$.

Proof of Theorem 4.5.

To simplify the notation we write $L^{k}$ instead of $L^{k}(\mbox{\it RC\/}_{\mathcal{K}})$ and $\gamma_{k}$ instead of $\gamma_{k}(\mbox{\it RC\/}_{\mathcal{K}})$. Statement (a) follows from the fact that

[TABLE]

with basis $\overline{g}_{1},\ldots,\overline{g}_{m}$.

We prove statement (b). Consider the abelianization map

[TABLE]

The group $\mbox{\it RC\/}_{\mathcal{K}}^{\prime}/\mbox{\it RC\/}_{\mathcal{K}}^{\prime\prime}=H_{1}(\mathcal{R}_{\mathcal{K}})$ is free abelian, see Corollary 2.4.

Consider $L^{2}=\gamma_{2}/\gamma_{3}$. The group $L^{2}$ is a $\mathbb{Z}_{2}$-module (see Proposition 3.3), i. e. $L^{2}=\mathbb{Z}_{2}^{M}$ for some $M\in\mathbb{N}$. We have a sequence of nested normal subgroups

[TABLE]

Consider the exact sequence of abelian groups:

[TABLE]

Recall from Corollary 2.4 that the free abelian group $\gamma_{2}/\gamma_{2}^{\prime}=\mathbb{Z}^{N}$ has a basis consisting of the images of the iterated commutators with all different indices described in Theorem 2.3. The images of the commutators of length $\geqslant 3$ are contained in the subgroup $\gamma_{3}/\gamma_{2}^{\prime}\subset\gamma_{2}/\gamma_{2}^{\prime}$. The group $\gamma_{3}/\gamma_{2}^{\prime}$ also contains commutators of length $3$ with duplicate indices, i. e. of the form $(g_{j},g_{i},g_{i})=(g_{i},g_{j})^{2}$. Therefore, the homomorphism $\psi$ acts by the formula:

[TABLE]

where the indices $i,j,k_{1},\ldots,k_{m-2}$ are all different. The elements $\overline{(g_{j},g_{i},g_{i})}$ with $i<j$, $\{i,j\}\notin\mathcal{K}$, and the elements $\overline{(g_{i},g_{j},g_{k_{1}},g_{k_{2}},\ldots,g_{k_{m-2}})},m\geqslant 3$, with the condition on the indices from Theorem 2.3 form a basis in a free abelian group $\gamma_{3}/\gamma_{2}^{\prime}$.

It follows that the $\mathbb{Z}_{2}$-module $L^{2}=\gamma_{2}/\gamma_{3}$ has a basis consisting of the elements $\overline{(g_{i},g_{j})}=[\overline{g}_{i},\overline{g}_{j}]$ with $i<j$ and ${\{i,j\}\notin\mathcal{K}}$, proving (b).

We prove statement (c). Consider $L^{3}=\gamma_{3}/\gamma_{4}$. The group $L^{3}$ is a $\mathbb{Z}_{2}$-module (see Proposition 3.3), i. e. $L^{3}=\mathbb{Z}_{2}^{M}$ for some $M\in\mathbb{N}$.

Consider the exact sequence of abelian groups:

[TABLE]

For the free abelian group $\gamma_{3}/\gamma_{2}^{\prime}$, we will use the basis constructed in the proof of statement (b). Elements of this basis corresponding to commutators of length $\geqslant 4$ are contained in $\gamma_{4}/\gamma_{2}^{\prime}$. The group $\gamma_{4}/\gamma_{2}^{\prime}$ also contains commutators of length $4$ with repeated indices. These commutators have one of the following nine types, which we divide into two types $A$ and $B$ for convenience:

[TABLE]

Note that

[TABLE]

because $(((g_{j},g_{i}),g_{j}),(g_{j},g_{i}))\in\gamma_{2}^{\prime}$. Here in the second identity we used commutator identity (2.3). A similar decomposition holds for other commutators of type $A$, for example,

[TABLE]

Now consider the commutators of type $B$. We will need the following commutator identities. For any $a,b,c,d\in\gamma_{1}$ we have:

[TABLE]

It follows that the last of the identities (2.3) takes the following form modulo $\gamma_{2}^{\prime}$:

[TABLE]

Furthermore, the following identity was obtained in [6, identity (4.5)]:

[TABLE]

If $x\in\gamma_{2}$, then the previous identity and identity (4.1) imply

[TABLE]

To simplify the notation, we write $i$ instead of $g_{i}$. From (2.3) and (4.2) we obtain

[TABLE]

[TABLE]

The last commutator of type $B$ requires a lengthier calculation:

[TABLE]

Here is the identity $\equiv^{1}$ is obtained with help of the algorithm written by the author in Wolfram Mathematica using commutator identities (2.3).

The identity $\equiv^{2}$ follows from the relations $(a,b)\cdot(a^{-1},b)=(b,a,a^{-1})$ and $(b,a,a^{-1})\equiv 1\mod\gamma_{2}^{\prime}$, if $a\in\gamma_{2}$.

The identity $\equiv^{3}$ follows from (4.3).

It follows that the homomorphism $\chi\colon\gamma_{4}/\gamma_{2}^{\prime}\rightarrow\gamma_{3}/\gamma_{2}^{\prime}$ acts by the formula:

[TABLE]

where the indices corresponding to a different letters are different. Thus, the $\mathbb{Z}_{2}$-module $L^{3}=\gamma_{3}/\gamma_{4}$ has a basis consisting of the elements specified in the theorem. ∎

As a consequence, we obtain a description of the first three consecutive quotients of the lower central series for a free product of the groups $\mathbb{Z}_{2}$.

Corollary 4.7**.**

Let $\mathcal{K}$ be a set of $m$ disjoint points, i. e. $\mbox{\it RC\/}_{\mathcal{K}}=\mathbb{Z}_{2}\langle g_{1}\rangle\ast\ldots\ast\mathbb{Z}_{2}\langle g_{m}\rangle$. Then:

• (a)

$L^{1}(\mbox{\it RC\/}_{\mathcal{K}})$* has a basis $\overline{g}_{1},\ldots,\overline{g}_{m}$;*

• (b)

$L^{2}(\mbox{\it RC\/}_{\mathcal{K}})$* has a basis consisting of the commutators $[\overline{g}_{i},\overline{g}_{j}]$ with $i<j$;*

• (c)

$L^{3}(\mbox{\it RC\/}_{\mathcal{K}})$* has a basis consisting of*

• –

the commutators $[\overline{g}_{i},\overline{g}_{j},\overline{g}_{j}]$ with $i<j$;

• –

the commutators $[\overline{g}_{i},\overline{g}_{j},\overline{g}_{k}]$ with $i<j>k$, $i\neq k$.