The space of marked Dyer systems, monotonicity, and continuity of growth rates

TL;DR

This paper studies the space of n-marked Dyer systems, showing it is closed, introduces a partial order making growth rates monotonic, and proves the growth rate function is continuous, extending Coxeter system results.

Contribution

It proves the closure of Dyer systems in the space of n-marked groups and establishes the continuity of growth rates within this class.

Findings

Dyer systems form a closed subspace in the space of n-marked groups.

A natural partial order on Dyer systems makes growth rates monotonic.

The growth rate function is continuous on the space of Dyer systems.

Abstract

The space $\mathcal{G}_n$ of $n$-marked groups provides a natural framework for studying algebraic and geometric invariants under deformation. In general, the growth rate is not continuous on $\mathcal{G}_n$. In this paper, we investigate the subspace $\mathcal{D}_n \subset \mathcal{G}_n$ consisting of $n$-marked Dyer systems, which extend Coxeter systems and include graph products of cyclic groups and right-angled Artin groups. We prove that $\mathcal{D}_n$ is closed in $\mathcal{G}_n$ and introduce a natural partial order on $\mathcal{D}_n$ with respect to which the growth rate is monotonically increasing. As a consequence, the growth rate function $\tau : \mathcal{D}_n \to \mathbb{R}_{\geq 1}$ is continuous. The proof combines the solution to the word problem for Dyer systems by Paris and Soergel, the parabolic growth formula by Paris and Varghese, and analytic arguments based on normal convergence and Hurwitz's theorem. This extends the continuity results known for Coxeter systems to the broader class of Dyer systems.