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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: BirkhĂ¤user (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)
##### Keywords:
surface; algebraic mapping-class group; unit-tangent bundle