Most big mapping class groups fail the Tits alternative

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