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Commensurators of surface braid groups. (English) Zbl 1245.20041

Given \(S_{g,p}\), the closed orientable surface of genus \(g\) and \(p\) boundary components, the pure braid group on \(p\) strands on \(\overline S_{g,p}\) is known to be isomorphic to the kernel of the homomorphism \(\iota\colon\text{PMod}(S_{g,p})\to\text{Mod}(S_{g,p})\), denoted by \(P(S)\), where \(\overline S_{g,p}\) denotes the closed surface obtained by attaching disks to all \(p\) boundary components of \(S_{g,p}\).
The automorphisms of \(PB_p(S_g)\) have been studied using techniques related to mapping class group, see for example P. Zhang [J. Knot Theory Ramifications 15, No. 9, 1231-1244 (2006; Zbl 1177.57016)]. The automorphisms of such groups are the automorphisms given by conjugation of elements of a larger group, namely \(\text{Mod}^*(S)\) \(\lambda\in B_p(S)\supset PB_p(S)\). The aim of this paper is to describe the automorphisms of any finite index subgroup of \(PB_p(S)\). The results are obtained using similar techniques to the case of \(PB_p(S)\). Also the same problem is considered for another group \(P_s(S)\), which is the subgroup of \(P(S)\) generated by all HBC twist and all HBP twist about separating HBP in \(S\).
The main result of this paper is: Theorem 1.1. Let \(S\) be a connected, compact and orientable surface of genus \(g\) with \(p\) boundary components. We assume \(g\geq 2\) and \(p\geq 2\). Then the following assertions hold:
(i) Let \(\Gamma_1\) and \(\Gamma_2\) be finite index subgroups of \(P(S)\), and let \(f\colon\Gamma_1\to\Gamma_2\) be an isomorphism. Then there exists an element \(\gamma\) of \(\text{Mod}^*(S)\) with \(f(x)=\gamma x\gamma^{-1}\) for any \(x\in\Gamma_1\).
(ii) Let \(\Lambda_1\) and \(\Lambda_2\) be finite index subgroups of \(P_s(S)\), and let \(h\colon\Lambda_1\to\Lambda_2\) be an isomorphism. Then there exists an element \(\lambda\) of \(\text{Mod}^*(S)\) with \(h(y)=\lambda y\lambda^{-1}\) for any \(y\in \Lambda_1\).
They make use of a complex which was defined motivated by the complex of curves. The vertices of that complex are given by either certain simple essential closed curves or pairs of simple curves called “hole-bounding pairs”. The study of this complex will provide the main result. Important steps are the computation of the dimension of the complex and the construction of the element \(\lambda\) which is quite long.

MSC:

20F36 Braid groups; Artin groups
20E36 Automorphisms of infinite groups
57M05 Fundamental group, presentations, free differential calculus
57M25 Knots and links in the \(3\)-sphere (MSC2010)
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)

Citations:

Zbl 1177.57016
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References:

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