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Algebraic mapping-class groups of orientable surfaces with boundaries. (English) Zbl 1176.57022
Bartholdi, Laurent (ed.) et al., Infinite groups: geometric, combinatorial and dynamical aspects. Based on the international conference on group theory: geometric, combinatorial and dynamical aspects of infinite groups, Gaeta, Italy, June 1–6, 2003. Basel: Birkhäuser (ISBN 3-7643-7446-2/hbk). Progress in Mathematics 248, 57-116 (2005).
Summary: Let \(S_{g,b,p}\) denote a surface which is connected, orientable, has genus \(g\), has \(b\) boundary components, and has \(p\) punctures. Let \(\Sigma_{g,b,p}\) denote the fundamental group of \(S_{g,b,p}\). Let \(\text{Out}_{g,b,p}\) denote the algebraic mapping-class group of \(S_{g,b,p}\).
We study the exact sequence
\[ 1\to \check\Sigma_{g,b,p}\to \text{Out}_{g,b\perp1,p}\to \text{Out}_{g,b,p}\to 1 \]
that arises from filling in the interior of a boundary component of \(S_{g,b+1,p}\). Here \(\text{Out}_{g,b\perp1,p}\) is a subgroup of index \(b+1\) in \(\text{Out}_{g,b+1,p}\). If \((g,b,p)\) is \((0,0,0)\) or \((0,0,1)\), then \(\check\Sigma_{g,b,p}\) is trivial. If \((g,b,p)\) is \((0,0,2)\) or \((1,0,0)\), then \(\check\Sigma_{g,b,p}\) is infinite cyclic. In all other cases, \(\check\Sigma_{g,b,p}\) is the fundamental group of the unit-tangent bundle of \(S_{g,b,p}\), a certain central extension of \(\Sigma_{g,b,p}\).
We give a description of the conjugation action of \(\text{Out}_{g,b\perp 1,p}\) on \(\check\Sigma_{g,b,p}\) in terms of the following three ingredients: the natural action of \(\text{Out}_{g,b\perp1,p}\) on \(\Sigma_{g,b+1,p}\); the natural homomorphism \(\Sigma_{g,b+1,p}\to\check\Sigma_{g,b,p}\); the twisting-number map \(\Sigma_{g,b+1,p}\to\mathbb Z\).
We apply our results to verify Matsumoto’s simplification of Wajnryb’s presentation of \(\text{Out}_{g,0,0}^+\).
For the entire collection see [Zbl 1083.20500].

MSC:
57M99 General low-dimensional topology
57M60 Group actions on manifolds and cell complexes in low dimensions
20F34 Fundamental groups and their automorphisms (group-theoretic aspects)
20E05 Free nonabelian groups
20E36 Automorphisms of infinite groups
20F36 Braid groups; Artin groups
58A30 Vector distributions (subbundles of the tangent bundles)
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