Automatic growth series for right-angled Coxeter groups

Abstract

Right-angled Coxeter groups have a natural automatic structure induced by their action on a CAT($0$) cube complex. The normal form for this structure is defined with respect to the generating set consisting of all cliques in the defining graph for the group. In this paper we study the growth series for right-angled Coxeter groups with respect to this automatic generating set. In particular, we show that there exist nonisomorphic Coxeter groups with the same automatic growth series, and give a comparison with the usual growth series defined with respect to the standard generating set.