The lower central and derived series of the braid groups of compact surfaces

TL;DR

This paper investigates the lower central and derived series of braid groups on compact surfaces, determining conditions for residual nilpotency and solubility, with explicit results for specific surfaces like the torus and Klein bottle.

Contribution

It provides new explicit calculations and general descriptions of the lower central and derived series for braid groups on various compact surfaces, including non-orientable cases.

Findings

B extsubscript{n}(M) and P extsubscript{n}(M) are residually nilpotent or soluble for specific n

Explicit lower central series for P extsubscript{2}(K) where K is the Klein bottle

Determined residual properties for braid groups on non-orientable surfaces

Abstract

Let M be a compact surface, either orientable or non-orientable. We study the lower central and derived series of the braid and pure braid groups of M in order to determine the values of n for which B\_n(M) and P\_n(M) are residually nilpotent or residually soluble. First, we solve this problem for the case where M is the 2-torus. We then give a general description of these series for an arbitrary semi-direct product that allows us to calculate explicitly the lower central series of P\_2(K), where K is the Klein bottle, and to give an estimate for the derived series of P\_n(K). Finally, if M is a non-orientable compact surface without boundary, we determine the values of n for which B\_n(M) is residually nilpotent or residually soluble in the cases that were not already known in the literature.

Equations

$$\{\begin{array}{rlr}& {\sigma }_i{\sigma }_{i+1}{\sigma }_i={\sigma }_{i+1}{\sigma }_i{\sigma }_{i+1}& \text{for all 1≤i≤n−2}\\ & {\sigma }_j{\sigma }_i={\sigma }_i{\sigma }_j& \text{if ∣i−j∣≥2 and 1≤i,j≤n−1.}\end{array}$$

$$F_n(M)=\left\{(x_1,\dots ,x_n):\text{xi∈M, and xi=xj if i=j, i,j=1,…,n}\right\}.$$

$$1⟶P_n(M)⟶B_n(M)⟶S_n⟶1.$$

$$1⟶P_m(M\setminus \left\{x_1,\dots ,x_n\right\})\stackrel{i_{\ast }}{⟶}{P}_{n+m}(M)\stackrel{p_{\ast }}{⟶}{P}_n(M)⟶1,$$

$$K_n$$

$${\tilde{H}}_n$$

$$V_n$$

$$\tilde{X}$$

$$\tilde{S}$$

$$\tilde{T}$$

$$B_n(\mathbb{T}{)}^{\text{Ab}}$$

$$B_n(\mathbb{K}{)}^{\text{Ab}}$$

$$1⟶H⟶G⟶K⟶1,$$

$$[x_{1},x_{2},\ldots,x_{n-1},x_{n}]=\Bigl{[}x_{1},\bigl{[}x_{2},\ldots,[x_{n-1},x_{n}]\bigr{]}\Bigr{]},$$

$$[x^{2^n},y]=[x,x,x^2,\dots ,x^{2^{n-1}},y].[x,x^2,\dots ,x^{2^{n-1}},y{]}^2.[x^2,\dots ,x^{2^{n-1}},y{]}^2\cdots [x^{2^{n-1}},y{]}^2.$$

$$[x^2,y]=x.x.y.x^{-1}.x^{-1}.y^{-1}=x[x,y]y{x}^{-1}{y}^{-1}=x[x,y]x^{-1}[x,y]=[x,x,y].[x,y{]}^2,$$

$$[x^{2^{n+1}},y]=[(x^{2^n}{)}^2,y]=[x^{2^n},x^{2^n},y][x^{2^n},y{]}^2,$$

$$\left[x^{2^n},[x^{2^n},y]\right]=\left[x,x,x^2,\dots ,x^{2^{n-1}},[x^{2^n},y]\right]{\left[x,x^2,\dots ,x^{2^{n-1}},[x^{2^n},y]\right]}^2\cdots {\left[x^{2^{n-1}},[x^{2^n},y]\right]}^2.$$

$$\bigl{[}x^{2^{n+1}},y\bigr{]}=\bigl{[}x,x,x^{2}\!,\ldots,x^{2^{n-1}}\!\!,[x^{2^{n}}\!\!,y]\bigr{]}\bigl{[}x,x^{2},\ldots,x^{2^{n-1}},[x^{2^{n}}\!\!,y]\bigr{]}^{2}\cdots\bigl{[}x^{2^{n-1}},[x^{2^{n}}\!\!,y]\bigr{]}^{2}\bigl{[}x^{2^{n}}\!\!,y\bigr{]}^{2},$$

$$hx{h}^{-1}=h(\phi (g)(y).y^{-1})h^{-1}=(\phi (g)(h^{\prime }y)y^{-1}{h}^{\prime -1})(\phi (g)(h^{\prime })h^{\prime -1}{)}^{-1}\in K_{n+1}\subset L_{n+1}.$$

$$hx{h}^{-1}$$

$$=(\phi (g)(h^{\prime }y{h}^{\prime -1}).(h^{\prime }y{h}^{\prime -1}{)}^{-1})[h^{\prime },y][y,\phi (g)(h^{\prime })]\in L_{n+1}$$

$$hx{h}^{-1}=[hy{h}^{-1},hl{h}^{-1}]\in [H,L_n]\subset L_{n+1}(\text{mbox}resp.[H,V_n]\subset V_{n+1}),$$

$$\phi (g)(x)=\phi (g)\left(\phi (g^{\prime })(h).h^{-1}\right)=\left(\phi (g{g}^{\prime })(h).h^{-1}\right){\left(\phi (g)(h).h^{-1}\right)}^{-1}.$$

$$\phi (g)(x)=[\phi (g)(h),\phi (g)(l)]\in [H,L_n],$$

$$y=\left(\phi (g^{-1}{g}^{\prime })(h).h^{-1}\right)h\left(\phi (g^{-1})(h^{-1}).h\right)h^{-1}\in L_{n+1},$$

$$[(h,g),(x,y)]=(h.\phi (g)(x).\phi (gy{g}^{-1})(h^{-1}).\phi ([g,y])(x^{-1}),[g,y]).$$

$$\rho =[h,x].xh{x}^{-1}\left(\phi (g)(x).x^{-1}\right)x{h}^{-1}{x}^{-1}.xh\left(\phi (gy{g}^{-1})(h^{-1}).h\right)h^{-1}{x}^{-1}.x\left(\phi ([g,y])(x^{-1}).x\right)x^{-1}.$$

$$(H{⋊}_{\phi }G{)}^{(n)}=[(H{⋊}_{\phi }G{)}^{(n-1)},(H{⋊}_{\phi }G{)}^{(n-1)}]\subset \left[V_n{⋊}_{\phi }{G}^{(n-1)},V_n{⋊}_{\phi }{G}^{(n-1)}\right].$$

$$\bigl{\langle}\varphi(g)(h).h^{-1}\,:\,g\in\widetilde{G},\,h\in\widetilde{H}\bigr{\rangle}$$

$$Z=\left\{\phi (g)(h).h^{-1}:g\in X,h\in Y\right\}$$

$$\left\{\phi (g)(w).w^{-1}:g\in X,w\in W\right\}.$$

$$L_{n+1}$$

Full text

1. Introduction

The braid groups of the disc, also called the Artin braid groups, were introduced by E. Artin [1]. If $n\geq 1$, the $n$-string Artin braid group, denoted by $B_{n}$, is generated by elements $\sigma_{1},\ldots,\sigma_{n-1}$ that are subject to the Artin relations:

[TABLE]

The notion of braid group was generalised to surfaces by Fox and Neuwirth using configuration spaces as follows [12]. Let $M$ be a compact, connected surface, and let $n\in\mathbb{N}$. The $n^{\text{th}}$* configuration space of $M$*, denoted by $F_{n}(M)$, is defined by:

[TABLE]

The $n$-string pure braid group $P_{n}(M)$ of $M$ is defined by $P_{n}(M)=\pi_{1}(F_{n}(M))$. The symmetric group $S_{n}$ on $n$ letters acts freely on $F_{n}(M)$ by permuting coordinates, and the $n$-string braid group $B_{n}(M)$ of $M$ is defined by $B_{n}(M)=\pi_{1}(F_{n}(M)/S_{n})$. This gives rise to the following short exact sequence:

[TABLE]

If $m\geq 1$, the projection $p\colon\thinspace F_{n+m}(M)\longrightarrow F_{n}(M)$ defined by $p(x_{1},\ldots,x_{n},\ldots,x_{n+m})=(x_{1},\ldots,x_{n})$ induces a homomorphism $p_{\ast}\colon\thinspace P_{n+m}(M)\longrightarrow P_{n}(M)$. Geometrically, $p_{\ast}$ is the homomorphism that ‘forgets’ the last $m$ strings. If $M$ is without boundary, Fadell and Neuwirth showed that $p$ is a locally-trivial fibration [9], with fibre $F_{m}(M\setminus\left\{x_{1},\ldots,x_{n}\right\})$ over the point $(x_{1},\ldots,x_{n})$, which we consider to be a subspace of the total space via the map $i\colon\thinspace F_{m}(M\setminus\left\{x_{1},\ldots,x_{n}\right\})\longrightarrow F_{n+m}(M)$ defined by $i((y_{1},\ldots,y_{m}))=(x_{1},\ldots,x_{n},y_{1},\ldots,y_{m})$. Applying the associated long exact sequence in homotopy to this fibration, we obtain the Fadell-Neuwirth short exact sequence of pure braid groups:

[TABLE]

where $n\geq 3$ if $M$ is the sphere $\mathbb{S}^{2}$ [8, 10], $n\geq 2$ if $M$ is the projective plane $\mathbb{R}P^{2}$ [10], and $n\geq 1$ otherwise [9], and $i_{\ast}$ is the homomorphism induced by the map $i$. This sequence has been widely studied. If $M$ is the torus $\mathbb{T}$ or the Klein bottle $\mathbb{K}$, the existence of a non-vanishing vector field on $M$ allows one to construct a section for $p$ [9]. This implies that the short exact sequence (1.2) splits for all $n,m\in\mathbb{N}$, and that $P_{n}(M)$ may be decomposed as an iterated semi-direct product (see Proposition 5.1 for an explicit section for $p_{\ast}$ in the case $M=\mathbb{K}$).

Let $G$ be a group. If $g,g^{\prime}\in G$ then $[g,g^{\prime}]=gg^{\prime}g^{-1}g^{\prime-1}$ denotes their commutator, and if $H$ and $K$ are subgroups of $G$, then the commutator subgroup of $H$ and $K$, denoted by $[H,K]$, is defined by [H,K]=\langle[h,k]:\text{h\in H$and$k\in K}\rangle, the subgroup of $G$ generated by the commutators of $H$ and $K$. The lower central series $\left\{\Gamma_{i}(G)\right\}_{i\geq 1}$ of $G$ is defined inductively by $\Gamma_{1}(G)=G$, and for $i\geq 1$, $\Gamma_{i+1}(G)=\left[\Gamma_{i}(G),G\right]$, and the derived series $\left\{G^{(i)}\right\}_{i\geq 0}$ of $G$ is defined inductively by $G^{(0)}=G$, and for $i\geq 0$, $G^{(i+1)}=\left[G^{(i)},G^{(i)}\right]$. The quotient $G/\Gamma_{2}(G)$ is the Abelianisation of $G$ that we denote by $G^{\text{Ab}}$. Following P. Hall, for any group-theoretic property $\mathcal{P}$, a group $G$ is said to be residually $\mathcal{P}$ if for any (non-trivial) element $x\in G$, there exists a group $H$ that possesses property $\mathcal{P}$ and a surjective homomorphism $\varphi:G\longrightarrow H$ such that $\varphi(x)\neq 1$. It is well known that a group $G$ is residually nilpotent (resp. residually soluble) if and only if $\bigcap_{i\geq 1}\Gamma_{i}(G)=\left\{1\right\}$ (resp. $\bigcap_{i\geq 0}G^{(i)}=\left\{1\right\}$).

If $p$ is a prime number, the lower $\mathbb{F}_{p}$-linear central filtration $\left\{\gamma^{p}_{i}(G)\right\}_{i\geq 1}$ of $G$ is defined inductively by $\gamma^{p}_{1}(G)=G$, and for $i\geq 1$, $\gamma^{p}_{i+1}(G)=\left\langle\left[\gamma^{p}_{i}(G),G\right],\,x^{p}\,:\,x\in\gamma^{p}_{i}(G)\right\rangle$ [27]. If the group $G$ is finitely generated, then $G$ is residually $p$-finite if and only if $\bigcap_{i\geq 1}\gamma^{p}_{i}(G)=\left\{1\right\}$ [27, Proposition 2.3(2)]. For any group $G$, $G^{(i)}\subset\Gamma_{i+1}(G)\subset\gamma^{p}_{i+1}(G)$, so residually $p$-finite implies residually nilpotent, which in turn implies residually soluble.

In this paper, we are interested in computing the lower central and derived series of $B_{n}(M)$ and $P_{n}(M)$, the aim being to determine the values of $n$ for which these groups are residually nilpotent or residually soluble. We also give some partial results for the residual $p$-finiteness of $P_{n}(\mathbb{K})$.

The first results about the lower central series of the braid groups were obtained by Gorin and Lin who gave a presentation of the commutator subgroup $\Gamma_{2}(B_{n})$ of $B_{n}$ for $n\geq 3$, and who showed that $(B_{n})^{(1)}=(B_{n})^{(2)}$ for all $n\geq 5$, which implies that $(B_{n})^{(1)}$ is perfect [20]. As a consequence, $\Gamma_{2}(B_{n})=\Gamma_{3}(B_{n})$, for all $n\geq 3$, so $B_{n}$ is not residually nilpotent. The lower central series of the pure braid group $P_{n}$ was studied by Falk and Randell [11] and by Kohno [24], who proved independently that $P_{n}$ is residually nilpotent for all $n\geq 1$.

The braid groups of orientable surfaces were studied by Bellingeri, Gervais and Guaschi [5]. If $M_{g,m}$ is a compact, connected, orientable surface of genus $g\geq 1$ with $m\geq 0$ boundary components, then $B_{n}(M_{g,m})$ is not residually nilpotent if $n\geq 3$, and $B_{2}(\mathbb{T})$ is residually nilpotent. In the case of the pure braid groups, $P_{n}(M_{g,m})$ is residually torsion-free nilpotent for all $n\geq 1$ if $m\geq 1$, or if $g=1$ and $m=0$ (the torus). If $m=0$ and $g\geq 1$, Bardakov and Bellingeri proved that $P_{n}(M_{g,m})$ is residually torsion-free nilpotent for all $n\geq 1$, and the braid group $B_{2}(M_{g,m})$ is residually $2$-finite, in particular, it is residually nilpotent [2]. Gonçalves and Guaschi studied the lower central and derived series of the braid groups of the sphere $\mathbb{S}^{2}$ and the projective plane $\mathbb{R}P^{2}$ [14, 17]. For the sphere, $B_{n}(\mathbb{S}^{2})$ is residually nilpotent if and only if $n\leq 2$, and residually soluble if and only if $n\leq 4$. In the case of the projective plane, $B_{n}(\mathbb{R}P^{2})$ is residually nilpotent if and only if $n\leq 3$, and if $n\neq 4$, $B_{n}(\mathbb{R}P^{2})$ is residually soluble if and only if $n<4$. More recently, if $M$ is a non-orientable surface different from $\mathbb{R}P^{2}$, Bellingeri and Gervais showed that $P_{n}(M)$ is residually $2$-finite, and so is residually nilpotent [4].

In this paper, we study the derived series of the torus and the lower central series and derived series of non-orientable surfaces. Our main results are as follows.

Theorem 1.1**.**

The group $B_{n}(\mathbb{T})$ is residually soluble if and only if $n\leq 5$.

For non-orientable surfaces, we first study the case of the Klein bottle. For an arbitrary semi-direct product, the following result describes its lower central series, and gives some information about its derived series. It will be used in the computation of the lower central and derived series of $P_{n}(\mathbb{K})$, but we believe that it may be applicable to other groups.

Theorem 1.2**.**

Let $G$ and $H$ be groups, and let $\varphi:G\longrightarrow Aut(H)$ be an action of $G$ on $H$. We define recursively the following subgroups of $H$: $L_{1}=V_{1}=H$, and if $n\geq 2$:

[TABLE]

Then $\varphi$ induces an action, which we also denote by $\varphi$, of $\Gamma_{n}(G)$ on $L_{n}$ (resp. of $G^{(n+1)}$ on $V_{n+2}$), and for all $n\in\mathbb{N}$, we have:

1. (1)

$\Gamma_{n}(H\rtimes_{\varphi}G)=L_{n}\rtimes_{\varphi}\Gamma_{n}(G)$. 2. (2)

$(H\rtimes_{\varphi}G)^{(n-1)}\subset V_{n}\rtimes_{\varphi}G^{(n-1)}$.

For the case of the commutator subgroup ($n=2$), part (1) was obtained in [14, Proposition 3.3]. Using Theorem 1.2, we compute explicitly $\Gamma_{n}(P_{2}(\mathbb{K}))$ and $\gamma^{2}_{n}(P_{2}(\mathbb{K}))$ in Theorems 5.4 and 5.13 respectively, and and it will follow that $P_{2}(\mathbb{K})$ is residually nilpotent and residually $2$-finite. If $n\geq 3$, in Theorem 5.20, we give an estimate of the derived series of $P_{n}(\mathbb{K})$, which will enable us to show that $P_{n}(\mathbb{K})$ is residually soluble, and will allow us to determine the values of $n$ for which $B_{n}(\mathbb{K})$ is residually nilpotent or residually soluble.

Theorem 1.3**.**

$B_{n}(\mathbb{K})$* is residually nilpotent if and only if $n\leq 2$, and residually soluble if and only if $n\leq 4$.*

For a non-orientable surface $M$ without boundary of higher genus, we may decide whether $B_{n}(M)$ is residually nilpotent or residually soluble using results of [4, 17].

Theorem 1.4**.**

Let $M$ be a compact non-orientable surface of genus $g$ without boundary. If $g\geq 2$ (resp. $g=1$), then $B_{n}(M)$ is residually nilpotent if and only if $n\leq 2$ (resp. $n\leq 3$). For all $g\geq 1$, $B_{n}(M)$ is residually soluble if and only if $n\leq 4$.

This paper is organised as follows. In Section 2, we give presentations of the braid groups used in this paper, as well as the statement of Theorem 2.5 due to Gruenberg that will be required in the proofs of some of our results. Theorem 1.2 is proved in Section 3. In Section 4, we study the case of the torus and we prove Theorem 1.1. In Section 5, our focus is on the braid groups of the Klein bottle, and we use Theorem 1.2 for the proof of Theorem 1.3. Theorem 1.4 is proved in Section 6. If $M$ is a compact surface different from $\mathbb{K}$ and the Möbius band, the centre $Z(B_{n}(M))$ of $B_{n}(M)$ is known [6, 7, 13, 26, 28, 31]. We determine $Z(B_{n}(\mathbb{K}))$ in Proposition 5.2, and for the sake of completeness, in Proposition A1 of the Appendix, we compute the centre of the braid groups of the Möbius band.

Acknowledgements

The authors would like to thank P. Bellingeri, S. Gervais, D. Gonçalves, L. Paris and D. Vendrúscolo for stimulating conversations. C. M. Pereiro was supported by project grant no 2010/18930-6 and 2012/01740-5 from FAPESP. During the writing of this paper, J. Guaschi was partially supported by the CNRS/FAPESP PRC project no 275209.

2. Generalities

In this section, we give the presentations of the braid and pure braid groups that will be used in this paper. If $M=\mathbb{T}$ or $\mathbb{K}$, we will make use of the following presentations of $P_{n}(M)$ and $B_{n}(M)$.

Theorem 2.1** ([29]).**

Let $n\geq 1$, and let $M$ be the torus $\mathbb{T}$ or the Klein bottle $\mathbb{K}$. The following constitutes a presentation of the pure braid group $P_{n}(M)$ of $M$:

generators: $\left\{a_{i},\,b_{i},\,i=1,\ldots,n\right\}\cup\left\{C_{i,j},\,1\leq i<j\leq n\right\}$.

relations:

1. (1)

$a_{i}a_{j}=a_{j}a_{i}$, $(1\leq i<j\leq n)$; 2. (2)

$a^{-1}_{i}b_{j}a_{i}=b_{j}a_{j}C^{-1}_{i,j}C_{i+1,j}a^{-1}_{j}$, $(1\leq i<j\leq n)$; 3. (3)

a^{-1}_{i}C_{j,k}a_{i}=\left\{\begin{array}[]{l}C_{j,k},\,\,(1\leq i<j<k\leq n)\,\,\mbox{or}\,\,(1\leq j<k<i\leq n),\\ a_{k}C^{-1}_{i+1,k}C_{i,k}a^{-1}_{k}C_{j,k}C^{-1}_{i,k}C_{i+1,k},\,\,(1\leq j\leq i<k\leq n);\end{array}\right.** 4. (4)

C^{-1}_{i,l}C_{j,k}C_{i,l}=\left\{\begin{array}[]{l}C_{j,k},\,\,(1\leq i<l<j<k\leq n)\,\,\mbox{or}\,\,(1\leq j\leq i<l<k\leq n),\\ C_{i,k}C^{-1}_{l+1,k}C_{l,k}C^{-1}_{i,k}C_{j,k}C^{-1}_{l,k}C_{l+1,k},\,\,(1\leq i<j\leq l<k\leq n);\end{array}\right.** 5. (5)

\left\{\begin{array}[]{lll}\prod^{n}_{j=i+1}C^{-1}_{i,j}C_{i+1,j}=a_{i}b_{i}C_{1,i}a^{-1}_{i}b^{-1}_{i},&(1\leq i\leq n),&\text{if}\,\,M=\mathbb{T},\\ \prod^{n}_{j=i+1}C_{i,j}C^{-1}_{i+1,j}=b_{i}C_{1,i}a^{-1}_{i}b^{-1}_{i}a^{-1}_{i},&(1\leq i\leq n),&\text{if}\,\,M=\mathbb{K};\end{array}\right.** 6. (6)

\left\{\begin{array}[]{ll}b_{j}b_{i}=b_{i}b_{j},\,\,(1\leq i<j\leq n),&\text{if}\,\,M=\mathbb{T},\\ b_{j}b_{i}=b_{i}b_{j}C_{i,j}C^{-1}_{i+1,j},\,\,(1\leq i<j\leq n),&\text{if}\,\,M=\mathbb{K};\end{array}\right.** 7. (7)

\left\{\begin{array}[]{ll}b^{-1}_{i}a_{j}b_{i}=a_{j}b_{j}C_{i,j}C^{-1}_{i+1,j}b^{-1}_{j},\,\,(1\leq i<j\leq n),&\text{if}\,\,M=\mathbb{T},\\ b^{-1}_{i}a_{j}b_{i}=a_{j}b_{j}(C_{i,j}C^{-1}_{i+1,j})^{-1}b^{-1}_{j},\,\,(1\leq i<j\leq n),&\text{if}\,\,M=\mathbb{K};\end{array}\right.** 8. (8)

\left\{\begin{array}[]{ll}{b^{-1}_{i}C_{j,k}b_{i}=\left\{\begin{array}[]{l}C_{j,k},\,\,(1\leq i<j<k\leq n)\,\,\mbox{or}\,\,(1\leq j<k<i\leq n),\\ C_{i+1,k}C^{-1}_{i,k}C_{j,k}b_{k}C_{i,k}C^{-1}_{i+1,k}b^{-1}_{k},\,\,(1\leq j\leq i<k\leq n);\end{array}\right.}&\text{if}\,\,M=\mathbb{T},\\ {b^{-1}_{i}C_{j,k}b_{i}=\left\{\begin{array}[]{l}C_{j,k},\,\,(1\leq i<j<k\leq n)\,\,\mbox{or}\,\,(1\leq j<k<i\leq n),\\ C_{i+1,k}C^{-1}_{i,k}C_{j,k}b_{k}(C_{i,k}C^{-1}_{i+1,k})^{-1}b^{-1}_{k},\,\,(1\leq j\leq i<k\leq n).\end{array}\right.}&\text{if}\,\,M=\mathbb{K}.\end{array}\right.**

Theorem 2.2** ([29]).**

Let $n\geq 1$, and let $M$ be the torus $\mathbb{T}$ or the Klein bottle $\mathbb{K}$. The following constitutes a presentation of the braid group $B_{n}(M)$ of $M$:

generators: $a,b,\sigma_{1},\ldots,\sigma_{n-1}$.

relations:

1. (1)

$\sigma_{i}\sigma_{i+1}\sigma_{i}=\sigma_{i+1}\sigma_{i}\sigma_{i+1}$; 2. (2)

$\sigma_{j}\sigma_{i}=\sigma_{i}\sigma_{j}$, if $|i-j|\geq 2$; 3. (3)

$a\sigma_{j}=\sigma_{j}a$, if $j\geq 2$; 4. (4)

$b\sigma_{j}=\sigma_{j}b$, if $j\geq 2$; 5. (5)

$b^{-1}\sigma_{1}a=\sigma_{1}a\sigma_{1}b^{-1}\sigma_{1}$; 6. (6)

$a(\sigma_{1}a\sigma_{1})=(\sigma_{1}a\sigma_{1})a$; 7. (7)

\left\{\begin{array}[]{lr}b(\sigma^{-1}_{1}b\sigma^{-1}_{1})=(\sigma^{-1}_{1}b\sigma^{-1}_{1})b,&\text{if}\,\,M=\mathbb{T},\\ b(\sigma^{-1}_{1}b\sigma_{1})=(\sigma^{-1}_{1}b\sigma^{-1}_{1})b,&\text{if}\,\,M=\mathbb{K};\end{array}\right.** 8. (8)

\sigma_{1}\sigma_{2}\cdots\sigma_{n-2}\sigma^{2}_{n-1}\sigma_{n-2}\cdots\sigma_{2}\sigma_{1}=\left\{\begin{array}[]{lr}bab^{-1}a^{-1}&\text{if}\,\,M=\mathbb{T},\\ ba^{-1}b^{-1}a^{-1}&\,\,\text{if}\,\,M=\mathbb{K}.\end{array}\right.**

We consider the torus and the Klein bottle to be a square whose edges are identified as indicated in Figure 1. Geometric representatives of the generators of $P_{n}(\mathbb{T})$ and $P_{n}(\mathbb{K})$ given in Theorem 2.1 are illustrated in Figure 2, and may be interpreted as follows. For $1\leq i\leq n$, the $i^{\text{th}}$ string is the only non-trivial string of the braid $a_{i}$ (resp. of $b_{i}$), and it passes through the edge $\alpha$ (resp. $\beta$). If $1\leq i<j\leq n$, the $j^{\text{th}}$ string is the only non-trivial string of the braid $C_{i,j}$, and it encircles all of the basepoints between the $i^{\text{th}}$ and $j^{\text{th}}$ points. If $i=j$, it will be convenient to define $C_{i,i}$ to be the trivial braid. The figures represent the projection of the braids onto $M$, so the constant paths in each figure correspond to vertical strings of the braid.

The generators of $B_{n}(\mathbb{T})$ and $B_{n}(\mathbb{K})$ given in Theorem 2.2 may be taken to be the standard Artin generators $\sigma_{1},\ldots,\sigma_{n-1}$ of $B_{n}$ as shown in Figure 3, and $a=a_{1}$ and $b=b_{1}$.

Various presentations of the braid and pure braid groups of the torus and the Klein bottle may be found in the literature [3, 6, 19, 30], but we choose to work with those of Theorems 2.1 and 2.2 because they highlight the similarities and differences between the braid groups of $\mathbb{T}$ and $\mathbb{K}$. For example, the word $C_{i,j}C^{-1}_{i+1,j}$ that appears in our presentation of $P_{n}(\mathbb{T})$ is often replaced by its inverse in $P_{n}(\mathbb{K})$. To prove Theorem 2.1 (resp. Theorem 2.2), one may use the Fadell-Neuwirth short exact sequence (1.2) (resp. the short exact sequence (1.1)), induction on $n$, and the following standard method for obtaining a presentation of a group extension [23, Proposition 1, p. 139]. Given a short exact sequence $1\longrightarrow A\stackrel{{\scriptstyle i}}{{\longrightarrow}}B\stackrel{{\scriptstyle p}}{{\longrightarrow}}C\longrightarrow 1$ and presentations $C=\left\langle X\,|\,R\right\rangle$ and $A=\left\langle Y\,|\,S\right\rangle$, then B=\bigl{\langle}\widetilde{X},\widetilde{Y}\,|\,\widetilde{S},\widetilde{R},\widetilde{T}\bigr{\rangle}, where:

[TABLE]

The details of the proofs of Theorems 2.1 and 2.2 are left to the reader.

Remark 2.3*.*

Using Theorem 2.2, it is straightforward to check that:

[TABLE]

for all $n\geq 2$, where $\overline{a}$ (resp. $\overline{b}$, $\sigma$) represents the $\Gamma_{2}$-coset of $a$ (resp. of $b$, $\sigma_{1}$).

For compact non-orientable surfaces of genus $g\geq 3$ without boundary, we shall make use of the following presentation of their braid groups due to Bellingeri.

Theorem 2.4** ([3]).**

Let $N_{g}$ be a compact, connected non-orientable surface of genus $g\geq 3$ without boundary. The braid group $B_{n}(N_{g})$ admits the following presentation:

generators: $\sigma_{1},\ldots,\sigma_{n-1},a_{1},\ldots,a_{g}$.

relations:

1. (1)

$\sigma_{i}\sigma_{i+1}\sigma_{i}=\sigma_{i+1}\sigma_{i}\sigma_{i+1}$; 2. (2)

$\sigma_{j}\sigma_{i}=\sigma_{i}\sigma_{j}$, if $|i-j|\geq 2$; 3. (3)

$a_{r}\sigma_{i}=\sigma_{i}a_{r}\,(1\leq r\leq g;\,i\neq 1)$; 4. (4)

$\sigma^{-1}_{1}a_{r}\sigma^{-1}_{1}a_{r}=a_{r}\sigma^{-1}_{1}a_{r}\sigma_{1}\,(1\leq r\leq g)$; 5. (5)

$\sigma^{-1}_{1}a_{s}\sigma_{1}a_{r}=a_{r}\sigma^{-1}_{1}a_{s}\sigma_{1}\,(1\leq s<r\leq g)$; 6. (6)

$a^{2}_{1}\cdots a^{2}_{g}=\sigma_{1}\sigma_{2}\cdots\sigma^{2}_{n-1}\cdots\sigma_{2}\sigma_{1}$.

To prove some of our results, we will also require the following theorem of Gruenberg.

Theorem 2.5** ([21]).**

Let $\mathcal{P}$ denote one of the following classes:

1. (1)

the class of soluble groups; 2. (2)

the class of finite groups; 3. (3)

the class of $p$-finite groups for a given prime number $p$.

Let $K$ and $H$ be groups, and suppose that $K$ is $\mathcal{P}$ and that $H$ is residually $\mathcal{P}$. Then, for any group extension

[TABLE]

the group $G$ is residually $\mathcal{P}$.

3. The lower central and derived series of semi-direct products

The main aim of this section is to establish the general decomposition of the lower central series and an estimate of the derived series of an arbitrary semi-direct product given in the statement of Theorem 1.2, which will be used in later computations of the lower central and derived series of $P_{n}(\mathbb{K})$. We first prove two lemmas that will be used in what follows. If $x_{1},\ldots,x_{n}$ are elements of a group $G$, we set:

[TABLE]

and if $X$ is a subset of $G$ then we denote the normal closure of $X$ in $G$ by $\left\langle\!\left\langle X\right\rangle\!\right\rangle_{G}$.

Lemma 3.1**.**

Let $G$ be a group, and let $x,y\in G$. For all $n\in\mathbb{N}$, we have:

[TABLE]

Proof.

We prove the lemma by induction on $n$. Observe that:

[TABLE]

which proves (3.1) in the case $n=1$. Now let $n\geq 2$, and suppose that the result holds for all $1\leq i\leq n$. Applying (3.2) to the elements $x^{2^{n}}$ and $[x^{2^{n}},y]$, we have:

[TABLE]

and applying (3.2) to the elements $x^{2^{n}}$ and $y$, we obtain:

[TABLE]

Thus:

[TABLE]

which completes the proof by induction. ∎

Remark 3.2*.*

With the notation of Theorem 1.2, In what follows, for the groups $K_{n}$, $H_{n}$ or $\widetilde{H}_{n}$, we will use the word generator to mean a word of the form $\varphi(g)(h).h^{-1}$, where $g\in\Gamma_{n-1}(G)$ and $h\in H$, $g\in G$ and $h\in V_{n-1}$, or $g\in\Gamma_{n-1}(G)$ and $h\in H$ respectively. Similarly, a generator of the group $L_{n}$ (resp. $V_{n}$) will mean either a generator of $K_{n}$ or $H_{n}$, or an element of the form $[h,l]$, where $h\in H$ and $l\in L_{n-1}$ (resp. either a generator of $\widetilde{H}_{n}$, or an element of the form $[h,v]$, where $h\in H$ and $v\in V_{n-1}$)

Lemma 3.3**.**

Let $n\geq 2$. With the notation of Theorem 1.2, the subgroups $K_{n}$, $L_{n}$ and $V_{n}$ are normal in $H$ for all $n\geq 2$, and we have the inclusions $K_{n+1}\subset K_{n}$, $H_{n+1}\subset H_{n}$, $\widetilde{H}_{n+1}\subset\widetilde{H}_{n}$, $L_{n+1}\subset L_{n}$ and $V_{n+1}\subset V_{n}$.

Proof.

The proof is by induction on $n$. The proof in the case $n=2$ was given in [14, Proposition 3.3]. So suppose that $n\geq 2$, and assume that $L_{n}$ (resp. $V_{n}$) is a normal subgroup of $H$, let $x\in L_{n+1}$ (resp. $V_{n+1}$) and let $h\in H$. Since L_{n+1}=\bigl{\langle}K_{n+1},H_{n+1},[H,L_{n}]\bigr{\rangle} (resp. V_{n+1}=\bigl{\langle}\widetilde{H}_{n+1},[H,V_{n}]\bigr{\rangle}), it suffices to show that $hxh^{-1}\in L_{n+1}$, where $x$ is a generator of $K_{n+1},H_{n+1}$ or $[H,L_{n}]$ (resp. of $\widetilde{H}_{n+1}$ or $[H,V_{n}]$), as in Remark 3.2.

• •

Suppose that $x=\varphi(g)(y)y^{-1}\in K_{n+1}$, where $g\in\Gamma_{n}(G)$ and $y\in H$. Then $\varphi(g)\in\operatorname{\text{Aut}}(H)$, and there exists $h^{\prime}\in H$ such that $\varphi(g)(h^{\prime})=h$, so:

[TABLE]

This also implies that $K_{n}$ is a normal subgroup of $H$ for all $n\geq 2$.

• •

Suppose that $x=\varphi(g)(y)y^{-1}$ is an element of $H_{n+1}$ (resp. of $\widetilde{H}_{n+1}$), where $g\in G$ and $y\in L_{n}$ (resp. $y\in V_{n}$), and let $h^{\prime}\in H$ be such that $\varphi(g)(h^{\prime})=h$. Then:

[TABLE]

because $h^{\prime}yh^{\prime-1}\in L_{n}$ (resp. $V_{n}$) by the normality of $L_{n}$ (resp. $V_{n}$) in $H$ using the induction hypothesis.

• •

Suppose that $x=[y,l]\in[H,L_{n}]$ (resp. $[H,V_{n}]$), where $y\in H$ and $l\in L_{n}$ (resp. $l\in V_{n}$). Then:

[TABLE]

because $hlh^{-1}\in L_{n}$ (resp. $V_{n}$) by the normality of $L_{n}$ (resp. $V_{n}$) in $H$.

This proves that $L_{n}$ (resp. $V_{n}$) is a normal subgroup of $H$ for all $n\geq 2$.

To prove the second part, notice that the inclusion $\Gamma_{n}(G)\subset\Gamma_{n-1}(G)$ implies that $K_{n+1}\subset K_{n}$ for all $n\geq 2$. It is easy to see that $H_{3}\subset H_{2}$ (resp. $\widetilde{H}_{3}\subset\widetilde{H}_{2}$) because $L_{2}\subset H$ (resp. $V_{2}\subset H$). By induction, suppose that $H_{n}\subset H_{n-1}$ (resp. $\widetilde{H}_{n}\subset\widetilde{H}_{n-1}$) for some $n\geq 3$. Since $L_{n-1}$ (resp. $V_{n-1}$) is normal in $H$, we have $[H,L_{n-1}]\subset L_{n-1}$ (resp. $[H,V_{n-1}]\subset V_{n-1}$). Further, using the definitions and the induction hypothesis, we have the inclusions $K_{n}\subset K_{n-1}\subset L_{n-1}$ and $H_{n}\subset H_{n-1}\subset L_{n-1}$ (resp. $\widetilde{H}_{n}\subset\widetilde{H}_{n-1}\subset V_{n-1}$). It follows that $L_{n}\subset L_{n-1}$ (resp. $V_{n}\subset V_{n-1}$), and then that $H_{n+1}\subset H_{n}$ (resp. $\widetilde{H}_{n+1}\subset\widetilde{H}_{n}$). Consequently, $L_{n+1}\subset L_{n}$ and $V_{n+1}\subset V_{n}$ for all $n\geq 2$. ∎

Proof of Theorem 1.2.

The proof is by induction on $n$. The case $n=1$ is trivial. If $n=2$, part (1) was proved in [14, Proposition 3.3], and part (2) follows from part (1) and the fact that $L_{2}=V_{2}$. Now suppose that parts (1) and (2) hold for some $n\geq 2$, and let us prove the result for $n+1$. Let $\varphi\colon\thinspace\Gamma_{n}(G)\longrightarrow\operatorname{\text{Aut}}(L_{n})$ be the action (also denoted by $\varphi$) induced by $\varphi$ such that $L_{n}\rtimes_{\varphi}\Gamma_{n}(G)=\Gamma_{n}(H\rtimes_{\varphi}G)$. We claim that $\varphi$ also induces an action $\varphi\colon\thinspace\Gamma_{n+1}(G)\longrightarrow\operatorname{\text{Aut}}(L_{n+1})$. To see this, let $g\in\Gamma_{n+1}(G)$. To prove that $\varphi(g)(L_{n+1})\subset L_{n+1}$, it suffices to take $x\in L_{n+1}$ to be of the form $x=\varphi(g^{\prime})(h).h^{-1}$, where either $g^{\prime}\in\Gamma_{n}(G)$ and $h\in H$, or $g^{\prime}\in G$ and $h\in L_{n}$, or of the form $x=[h,l]\in[H,L_{n}]$, where $h\in H$ and $l\in L_{n}$. The result will then follow for all elements of $L_{n+1}$ because $\varphi(g)$ is a homomorphism.

• •

If $x=\varphi(g^{\prime})(h).h^{-1}\in K_{n+1}$, where $g^{\prime}\in\Gamma_{n}(G)$ and $h\in H$, or $x=\varphi(g^{\prime})(h).h^{-1}\in H_{n+1}$, where $g^{\prime}\in G$ and $h\in L_{n}$, then:

[TABLE]

If $g^{\prime}\in\Gamma_{n}(G)$ and $h\in H$ then $\varphi(g)(x)\in K_{n+1}\subset L_{n+1}$ since $g$ and $gg^{\prime}$ belong to $\Gamma_{n}(G)$. If $g^{\prime}\in G$ and $h\in L_{n}$, then $\varphi(g)(x)\in H_{n+1}\subset L_{n+1}$ because $h\in L_{n+1}$.

• •

If $x=[h,l]\in[H,L_{n}]$, where $h\in H$ and $l\in L_{n}$ then:

[TABLE]

since $g\in\Gamma_{n+1}(G)\subset\Gamma_{n}(G)$, so $\varphi(g)(l)\in L_{n}$.

Since $\varphi(g)\colon\thinspace L_{n+1}\longrightarrow L_{n+1}$ is the restriction of an automorphism, it is injective, so to show that it is an automorphism, it suffices to prove surjectivity. We first consider the following two cases:

1. (a)

If $x=\varphi(g^{\prime})(h).h^{-1}$, where either $g^{\prime}\in\Gamma_{n}(G)$ and $h\in H$, or $g^{\prime}\in G$ and $h\in L_{n}$, let:

[TABLE]

because $L_{n+1}\triangleleft H$, and one may check that $\varphi(g)(y)=x$. 2. (b)

If $x=[h,l]\in[H,L_{n}]$, where $h\in H$ and $l\in L_{n}$, there exist $l^{\prime}\in L_{n}$ and $h^{\prime}\in H$ such that $\varphi(l^{\prime})=l$ and $\varphi(g)(h^{\prime})=h$ by the induction hypothesis. Taking $y=[h^{}$