#### 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
##### 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, $Map\left(X\right)$ contains a finitely generated subgroup that is not virtually solvable and contains no nonabelian free group.

##### Keywords
mapping class group, Grigorchuk group, Tits alternative
Primary: 57K20
Secondary: 20F38