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Geometric subgroups of surface braid groups. (English) Zbl 0962.20028

Let \(H\) be a subgroup of a group \(G\). Recall that the commensurator \(C_G(H)\) of \(H\) in \(G\) is the subgroup of all \(g\in G\) such that \(H\cap gHg^{-1}\) has finite index in both \(H\) and \(gHg^{-1}\). In [Invent. Math. 130, No. 3, 575-587 (1997; Zbl 0897.20031)], the second author determined the commensurators, normalizers, and centralizers of the natural braid subgroups \(B_n\) in \(B_m\) (\(n\leq m\)), and used these results to show (for example) that certain induced representations of \(B_m\) are irreducible.
Motivated by such applications, and by the importance of surface braid groups in, for example, quantum field theory, in this paper, the authors obtain analogous results for subgroups of a braid group on a surface \(M\) coming from braid groups on certain natural subsurfaces \(N\subseteq M\). For example, it is shown that if \(M\) is large (fundamental group not virtually Abelian), \(N\) is neither a disk nor the collar of a Möbius strip in \(M\), and no component of \(\overline{M\setminus N}\) is a disk, then, roughly speaking, the commensurator of a braid group on \(N\) is the direct product of that braid group and braid groups on the components of \(\overline{M\setminus N}\). The techniques used include amalgams, various descriptions of braid groups in terms of exact sequences (i.e., as extensions), and an analysis of the action of \(\pi_1(N)\) on the fundamental groupoid of \(M\setminus P_0\) for \(P_0\in N\).

MSC:

20F36 Braid groups; Artin groups
57M05 Fundamental group, presentations, free differential calculus
57M07 Topological methods in group theory
57M60 Group actions on manifolds and cell complexes in low dimensions
20E07 Subgroup theorems; subgroup growth

Citations:

Zbl 0897.20031
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References:

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