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

TL;DR

This paper proves that the growth rate of Coxeter systems varies continuously within their space, extending previous results from hyperbolic polygons and polyhedra to a broader group-theoretic context.

Contribution

It establishes the continuity of growth rates for Coxeter systems, generalizing Floyd and Kolpakov's findings from geometric polyhedra to abstract Coxeter groups.

Findings

Growth rate is a continuous function on the space of Coxeter systems.

Extends geometric convergence results to algebraic group settings.

Provides a unified framework for understanding growth rate behavior.

Abstract

Floyd showed that if a sequence of compact hyperbolic Coxeter polygons converges, then so does the sequence of the growth rates of the Coxeter groups associated with the polygons. For the case of the hyperbolic 3-space, Kolpakov discovered the same phenomena 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 systems. 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 systems in the space of marked groups.