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Geometric and Functional Analysis

, Volume 22, Issue 6, pp 1541–1590 | Cite as

Right-angled Artin groups and a generalized isomorphism problem for finitely generated subgroups of mapping class groups

  • Thomas KoberdaEmail author
Article

Abstract

Consider the mapping class group Mod g,p of a surface Σ g,p of genus g with p punctures, and a finite collection {f1, . . . , fk} of mapping classes, each of which is either a Dehn twist about a simple closed curve or a pseudo-Anosov homeomorphism supported on a connected subsurface. In this paper we prove that for all sufficiently large N, the mapping classes \({\{f_1^N,\ldots,f_k^N\}}\) generate a right-angled Artin group. The right-angled Artin group which they generate can be determined from the combinatorial topology of the mapping classes themselves. When {f1, . . . , fk} are arbitrary mapping classes, we show that sufficiently large powers of these mapping classes generate a group which embeds in a right-angled Artin group in a controlled way. We establish some analogous results for real and complex hyperbolic manifolds. We also discuss the unsolvability of the isomorphism problem for finitely generated subgroups of Mod g,p , and recover the fact that the isomorphism problem for right-angled Artin groups is solvable. We thus characterize the isomorphism type of many naturally occurring subgroups of Mod g,p .

Keywords and phrases

Subgroups of mapping class groups right-angled Artin groups 

Mathematics Subject Classification (1991)

Primary 37E30 Secondary 20F36 05C60 

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Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  1. 1.Department of MathematicsYale UniversityNew HavenUSA

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