Exponential growth rates of even Coxeter groups

Abstract: Let $W$ be an even Coxeter group. We prove that among all Coxeter systems generating $W$ the unique even Coxeter system realizes the minimal exponential growth. Our proof relies on comparing the exponential growth rates in the explicit algorithm of Mihalik which from any Coxeter system of an even Coxeter group eventually produces the unique even one. The main new ingredient is that blow downs along pseudo-transpositions do not increase the exponential growth rate.

• The paper proves that the minimal exponential growth rate of even Coxeter groups is uniquely achieved using the standard even generating set.
• It demonstrates that performing blow downs along pseudo-transpositions does not increase the growth rate, with explicit numerical computations supporting the claim.
• The findings offer algorithmic reduction methods and insights into rigidity and growth gaps in Coxeter groups, impacting both theoretical and computational group studies.

Exponential Growth Rates in Even Coxeter Groups

Introduction

The paper "Exponential growth rates of even Coxeter groups" (2607.01879) studies the exponential and uniform exponential growth rates for even Coxeter groups, focusing on the relationship between these rates as computed over the set of all Coxeter generating sets. The primary achievement is the proof that for any even Coxeter group $W$, the minimal exponential growth rate, when restricted to Coxeter generating sets, is realized uniquely at the standard even generating set. A methodological contribution is the analysis showing that certain group-theoretic operations ("blow downs" along pseudo-transpositions) do not increase the growth rate, consolidating the identification of the growth-minimizing generating set via an algorithmic reduction process.

Background and Notation

For a finitely generated group $G$ and generating set $S$, the exponential growth rate $\omega(G,S)$ is defined as:

$$\omega(G, S) = \limsup_{n\to\infty} \sqrt[n]{|\{g \in G: \ell_S(g) = n\}|}$$

where $\ell_S(g)$ is the word length with respect to $S$. The uniform exponential growth rate is then $\omega(G) = \inf_S \omega(G, S)$, infimum taken over all finite generating sets.

In the context of Coxeter groups, one may restrict attention to Coxeter generating sets, yielding

$$\omega_{\mathrm{Cox}}(W) := \inf_{S \text{ Coxeter}} \omega(W, S)$$

An even Coxeter group is defined as one admitting a Coxeter system $(W, S_{\mathrm{even}})$ such that all nontrivial edge labels in its Coxeter diagram are even or infinite; crucially, such a system is unique (Bahls [Bahls], Mihalik [Mihalik]).

Main Results

Minimal Growth Realization

The central theorem established in the paper is:

Let $G$0 be an even Coxeter group. Then the infimum of exponential growth rates over Coxeter generating sets is achieved uniquely at the even Coxeter system, i.e., $G$1 Here, $G$2 denotes the unique even Coxeter generating set.

This is accomplished by meticulously analyzing Mihalik's reduction algorithm, which transforms any Coxeter system for an even Coxeter group into the unique even Coxeter system by a sequence of diagram twisting and "blow downs" along pseudo-transpositions, each step either preserving or reducing the growth rate.

Non-increasing Growth via Blow Downs

A pivotal technical innovation is the proof that blow downs along pseudo-transpositions—that is, replacing certain generating reflections according to explicit group-theoretic rules—cannot increase the exponential growth rate:

For a Coxeter system $G$3 and a pseudo-transposition $G$4, if $G$5 is the new generating set after blow down (with precise elements described in the paper), then $G$6 The proof leverages the structure of Coxeter groups and Steinberg's formula for growth series, reducing the analysis to positivity properties of certain rational functions under substitution, and showing strict inequality in broad classes of hyperbolic triangular groups.

Examples and Numerical Results

The authors provide explicit computations for classes of Coxeter groups including triangular groups, amalgamated products, and free products with finite groups. Notably:

• In certain hyperbolic triangular groups, the strict inequality $G$7 holds, with numerical values computed for small parameters (e.g., for $G$8, $G$9: $S$0).
• For groups of the form $S$1, the minimal growth rate over all generating sets is strictly less than over Coxeter systems, i.e.,

$S$2

This illustrates that restricting to Coxeter generating sets can yield a strictly larger "minimal" exponential growth rate.

• The analysis is supported by explicit calculations of Poincaré series and their radii of convergence, utilizing Steinberg's formula and other results on free and amalgamated products.

Theoretical and Practical Implications

The findings have several ramifications:

• Rigidity in Growth Minimization: Even Coxeter groups (including important special cases such as finite reflection groups and infinite Euclidean or hyperbolic reflection groups with even labeling) possess a canonical generating set minimizing growth, reaffirming the distinguished role of the standard even system in geometric and combinatorial group theory.
• Algorithmic Reduction: The result implies the practical feasibility of computing minimal Coxeter growth rates by algorithmic reduction to the even system, without needing to consider the vast class of arbitrary Coxeter generating sets.
• Growth Rate Discontinuities: The strict inequality examples confirm that $S$3 can exceed $S$4, which has implications for geometric group theory, particularly in understanding the behavior of growth invariants under restrictions to specific types of generating sets.
• Connections to Rigidity and Isomorphism Problems: The proof techniques exploit the rigidity of Coxeter diagrams, the uniqueness of even systems, and the structure of maximal parabolics, highlighting the deep interplays between growth, group presentations, and automorphisms.

Prospects for Further Study

Potential directions for future research include:

• Extension to Broader Classes: Investigating analogous growth minimization phenomena in Artin groups, complex reflection groups, or other classes with diagrammatic presentations.
• Refined Asymptotics: Developing precise asymptotic formulas for growth rates and series coefficients in infinite Coxeter groups, especially in the hyperbolic regime.
• Algorithmic Applications: Leveraging the reduction algorithms for computational group theory tasks involving enumeration, word problems, or spectral estimates of Cayley graphs.
• Growth Gap Analysis: Systematic exploration of the "growth gap" between $S$5 and $S$6 in various families of groups, possibly leading to new invariants or classification results.

Conclusion

The paper establishes that the unique even Coxeter system of an even Coxeter group realizes the minimal exponential growth rate among all Coxeter generating sets, with the reduction process via diagram twisting and pseudo-transpositions strictly non-increasing in growth. The findings clarify structural and algorithmic aspects of Coxeter groups relevant to both explicit computation and theoretical understanding of group growth, and open avenues for further inquiry into group-theoretic growth invariants.