The automorphism group of the universal Coxeter group

TL;DR

This paper investigates the fixed point properties and rigidity of the automorphism group of the universal Coxeter group, showing it acts trivially on certain CAT(0) spaces and lacks Kazhdan's property (T).

Contribution

It establishes new fixed point theorems for Aut(W_n) on CAT(0) spaces and proves the absence of Kazhdan's property (T) for these groups.

Findings

Aut(W_n) fixes a point in low-dimensional CAT(0) spaces

Aut(W_n) does not have Kazhdan's property (T)

Restrictions on homomorphisms to groups without Sym(n)

Abstract

We study fixed point properties of the automorphism group of the universal Coxeter group Aut$(W_n)$. In particular, we prove that whenever Aut$(W_n)$ acts by isometries on complete $d$-dimensional CAT$(0)$ space with $d<\lfloor\frac{n}{2}\rfloor$, then it must fix a point. We also prove that Aut$(W_n)$ does not have Kazhdan's property (T). Further, strong restrictions are obtained on homomorphisms of Aut$(W_n)$ to groups that do not contain a copy of Sym(n).