Most big mapping class groups fail the Tits alternative (Q2070110)

A big mapping class group is the mapping class group of a surface of infinite type, i.e. of a surface whose fundamental group is not finitely generated. A group satisfies the Tits alternative if every finitely generated subgroup is either ``big'' (contains a nonabelian free group) or ``small'' (contains a finite-index solvable group) (this was shown by Tits for subgroups of \(\mathrm{GL}_n(k)\)). It is known that mapping class groups of surfaces of finite type satisfy the Tits alternative; on the other hand, examples of big mapping class groups were known which do not satify the Tits alternative (e.g. mapping class groups that contain every countable group; an example is the mapping class group of the Loch Ness monster surface, see the recent preprint of \textit{Y. Krifka} and \textit{D. Spriano} [``A note on subgroups of the Loch Ness monster surface's mapping class group'', Preprint, \url{arXiv:2201.10378}] for a short proof). The main result of the present paper is the following. Let \(X\) be a surface, possibly nonorientable and with boundary; then the mapping class group does not satisfy the Tits alternative if \(X\) has infinite genus, or has infinitely many punctures, or contains as a closed subset a 2-disk with a Cantor set removed from its interior. As a corollary, every surface of infinite type with only finitely many boundary components does not satisfy the Tits alternative. ``The key idea in the proof of the main theorem is to construct \(\tilde G\) with a surjection \(\tilde G \to G\), where \(G\) is Grigorchuk's group (a group of automorphisms of a rooted binary tree). We do this in such a way that the kernel is abelian, which allows us to tranfer to \(\tilde G\) the fact that \(G\) does not satisfy the Tits alternative.''