# Centers of subgroups of big mapping class groups and the Tits alternative

## TL;DR

This paper investigates the algebraic structure of large subgroups of big mapping class groups of infinite-type surfaces, revealing trivial centers, the presence of free groups, and the failure of the Tits alternative.

## Contribution

It demonstrates that many subgroups have trivial centers, all nontrivial normal subgroups contain free groups, and big mapping class groups do not satisfy the Tits alternative.

## Key findings

- Many subgroups of big mapping class groups have trivial centers.
- Every nontrivial normal subgroup contains a nonabelian free group.
- Big mapping class groups do not satisfy the strong Tits alternative.

## Abstract

In this note we show that many subgroups of mapping class groups of infinite-type surfaces without boundary have trivial centers, including all normal subgroups. Using similar techniques, we show that every nontrivial normal subgroup of a big mapping class group contains a nonabelian free group. In contrast, we show that no big mapping class group satisfies the strong Tits alternative enjoyed by finite-type mapping class groups. We also give examples of big mapping class groups that fail to satisfy even the classical Tits alternative and give a proof that every countable group appears as a subgroup of some big mapping class group.