Volume 21, issue 7 (2021)

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Most big mapping class groups fail the Tits alternative

Daniel Allcock

Algebraic & Geometric Topology 21 (2021) 3675–3688

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.

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mapping class group, Grigorchuk group, Tits alternative
Mathematical Subject Classification
Primary: 57K20
Secondary: 20F38
Received: 31 August 2020
Revised: 1 December 2020
Accepted: 24 December 2020
Published: 28 December 2021
Daniel Allcock
Department of Mathematics
University of Texas, Austin
Austin, TX
United States