Generalisations of the Tits Representation

Abstract

We construct a group $K_n$ with properties similar to infinite Coxeter groups. In particular, it has a geometric representation featuring hyperplanes, simplicial chambers and a Tits cone. The generators of $K_n$ are given by $2$-element subsets of $\{0,\ldots,n\}$. We provide some generalities to deal with groups like these. We give some easy combinatorial results on the finite residues of $K_n$, which are equivalent to certain simplicial real central hyperplane arrangements.