On presentations for mapping class groups of orientable surfaces via Poincaré's Polyhedron theorem and graphs of groups - Archive ouverte HAL Accéder directement au contenu
Pré-Publication, Document De Travail Année : 2021

On presentations for mapping class groups of orientable surfaces via Poincaré's Polyhedron theorem and graphs of groups

Résumé

The mapping class group of an orientable surface with one boundary component, S, is isomorphic to a subgroup of the automorphism group of the fundamental group of S. We call these subgroups algebraic mapping class groups. An algebraic mapping class group acts on a space called ordered Auter space. We apply Poincaré's Polyhedron theorem to this action. We describe a decomposition of ordered Auter space. From these results, we deduce that the algebraic mapping class group of S is a quotient of the fundamental group of a graph of groups with, at most, two vertices and, at most, six edges. Vertex and edge groups of our graph of groups are mapping class groups of orientable surfaces with one, two or three boundary components. A presentation for the mapping class group of S can be obtained by adding, at most, 24 relations to the fundamental group of our graph of groups.
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Dates et versions

hal-03100803 , version 1 (06-01-2021)

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  • HAL Id : hal-03100803 , version 1

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Lluís Bacardit. On presentations for mapping class groups of orientable surfaces via Poincaré's Polyhedron theorem and graphs of groups. 2021. ⟨hal-03100803⟩
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