Commensurations of the outer automorphism group of a universal Coxeter group

This paper studies the rigidity properties of the abstract commensurator of the outer automorphism group of a universal Coxeter group of rank n, which is the free product $W_n$ of $n$ copies of $\mathrm{Z}/2\mathrm{Z}$. We prove that for $n \neq 5$ the natural map $\mathrm{Out}(W_n) \to \mathrm{Comm}(\mathrm{Out}(W_n)$ is an isomorphism and that every isomorphism between finite index subgroups of $\mathrm{Out}(W_n)$ is given by a conjugation by an element of $\mathrm{Out}(W_n)$.