Commensurations of the outer automorphism group of a universal Coxeter group

TL;DR

This paper establishes the rigidity of the outer automorphism group of a universal Coxeter group by showing the natural map to its commensurator is an isomorphism for rank n ≥ 5, indicating strong structural stability.

Contribution

It proves the isomorphism between the outer automorphism group and its commensurator for universal Coxeter groups of rank at least 5, revealing new rigidity properties.

Findings

The natural map from Out(W_n) to its commensurator is an isomorphism for n ≥ 5.

Any isomorphism between finite index subgroups of Out(W_n) is realized by conjugation.

The results demonstrate strong rigidity in the structure of Out(W_n) for large n.

Abstract

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 $\mathbb{Z}/2\mathbb{Z}$. We prove that for $n\geq 5$ 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 $\mathrm{Out}(W_n)$.