The automorphism group of the universal Coxeter group

Olga Varghese

paper

The automorphism group of the universal Coxeter group

arXiv:1805.06748

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 $d < ⌊ \frac{n}{2} ⌋$ , 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).

6 pages