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

TL;DR

The paper studies the deformation space $\mathcal{G}_n$ of $n$-marked groups and the growth rate $\tau(G,S)$, focusing on the natural subspace of $n$-marked Dyer systems $\mathcal{D}_n$, which generalizes Coxeter systems. It proves that $\mathcal{D}_n$ is compact (closed in $\mathcal{G}_n$) and introduces a monotone partial order under which the growth rate is nondecreasing, yielding $\tau$-monotonicity. By combining the Paris–Soergel word problem for Dyer groups, the Paris–Varghese parabolic growth formula, and analytic tools (normal convergence and Hurwitz), the paper establishes the continuity of the growth rate function $\tau$ on $\mathcal{D}_n$, extending Coxeter continuity results to this broader class. The results rely on the tight interplay between the algebraic structure (Dyer graphs/matrices and their embeddings in Coxeter groups) and analytic properties of growth series. This provides a robust deformation-theoretic insight into growth behavior for a wide family that includes graph products of cyclic groups and right-angled Artin groups.

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 $τ: \mathcal{D}_n \to \mathbb{R}_{\geq 1}$ 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.

Figures (4)

• Figure 1: The Coxeter graphs of irreducible spherical Coxeter systems with $N$ vertices
• Figure 2: The Coxeter graphs of irreducible Euclidean Coxeter systems with $N+1$ vertices
• Figure 3: Example of Dyer graph $\Gamma$
• Figure 4: Example of induced Coxeter graph: the left-hand side is a Dyer graph $\Gamma$, and the right-hand side is the induced Coxeter graph $\Lambda$

Theorems & Definitions (38)

• Theorem A: Compactness of $\mathcal{D}_n$
• Theorem B: Monotonicity of growth rates
• Theorem C: Continuity of growth rates
• Example 2.1
• Lemma 2.1
• Lemma 2.2
• proof
• Example 2.2
• Definition 2.1
• Theorem 3.1: Yukita2024