Most big mapping class groups fail the Tits Alternative

TL;DR

This paper demonstrates that the mapping class groups of certain infinite-type surfaces do not satisfy the Tits Alternative, revealing complex subgroup structures that challenge previous assumptions about these groups.

Contribution

It proves that for a broad class of infinite-type surfaces, their mapping class groups contain finitely generated subgroups that are not virtually solvable and lack nonabelian free groups.

Findings

Mapping class groups of infinite-type surfaces do not satisfy the Tits Alternative.

Existence of finitely generated, non-virtually solvable subgroups without free groups.

Challenges previous beliefs about the structure of mapping class groups.

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.