Most big mapping class groups fail the Tits alternative

Allcock, Daniel

doi:10.2140/agt.2021.21.3675

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Most big mapping class groups fail the Tits alternative

Daniel Allcock

Algebraic & Geometric Topology 21 (2021) 3675–3688

DOI: 10.2140/agt.2021.21.3675

Abstract

Let $X$ be a surface, possibly with boundary. Suppose it has infinite genus or infinitely many punctures, or a closed subset which is a disk with a Cantor set removed from its interior. For example, $X$ could be any surface of infinite type with only finitely many boundary components. We prove that the mapping class group of $X$ does not satisfy the Tits alternative. That is, $Map (X)$ contains a finitely generated subgroup that is not virtually solvable and contains no nonabelian free group.

Keywords

mapping class group, Grigorchuk group, Tits alternative

Mathematical Subject Classification

Primary: 57K20

Secondary: 20F38

References

Publication

Received: 31 August 2020

Revised: 1 December 2020

Accepted: 24 December 2020

Published: 28 December 2021

Authors

Daniel Allcock
Department of Mathematics University of Texas, Austin Austin, TX United States
http://www.math.utexas.edu/~allcock

Volume 21, issue 7 (2021)