On the continuity of the growth rate on the space of Coxeter groups

Tomoshige Yukita

Published 2020-11-19Version 1

Floyd showed that if a sequence of compact hyperbolic Coxeter polygons converges, then so do the growth rates of the Coxeter groups associated with the polygons. For the case of the hyperbolic 3-space, Kolpakov discovered the same phenomenon for specific convergent sequences of hyperbolic Coxeter polyhedra. In this paper, we show that the growth rate is a continuous function on the space of Coxeter groups. This is an extension of the results due to Floyd and Kolpakov since the convergent sequences of Coxeter polyhedra give rise to that of Coxeter groups in the space of marked groups.