# '22

36th Summer Topology Conference

July 18-22, 2022

University of Vienna, Department of Mathematics
Oskar-Morgenstern-Platz 1, 1090 Vienna, AUSTRIA

### "The automorphism group of the free group of rank two and the five-punctured sphere."

The abstract commensurator of a group $G$ is the group of isomorphisms between finite-index subgroups of $G$, where two isomorphisms are considered equal if they agree on a common finite-index subgroup of their domains. We shall briefly survey existing results about abstract commensurators of automorphism groups of free groups and give geometric reasoning behind the following, initially surprising, fact: the abstract commensurator of the automorphism group of a free group of rank two is equal to the extended mapping class group of the five-punctured sphere. This is based on forthcoming work with Martin Bridson.