Stable reflection length in Coxeter groups

TL;DR

This paper introduces stable reflection length $\mathrm{srl}$ in Coxeter groups and establishes its deep connections to stable commutator length $\mathrm{scl}$ and stable torsion length $\mathrm{stl}$, including a bi-Lipschitz relation between $\mathrm{stl}$ and $\mathrm{srl}$ and a full combinatorial criterion for when $\{\mathrm{rl}(w^n)\}$ is unbounded via the straight part of $w$ and its parabolic closure. It proves a key positivity result: for irreducible indefinite type with $\mathrm{Pc}(w)=W$, $\mathrm{scl}_W(w)>0$ if and only if $w$ is chiral, bridging to geometry of group actions and homogeneous quasimorphisms. Chirality is then characterized in terms of products of involutions, showing that achirality forces the straight part to be a product of two involutions and leading to a suite of equivalent conditions involving $\mathrm{rl}$-growth and $\mathrm{srl}$/$\mathrm{scl}$. The Coxeter-element analysis yields precise graph-theoretic criteria: $\Gamma$ bipartite iff some Coxeter element is conjugate to its inverse, and $\Gamma$ a tree iff all Coxeter elements are, with the boundedness of reflection length along powers tied to these structural properties.

Abstract

We introduce stable reflection length in Coxeter groups, as a way to study the asymptotic behaviour of reflection length. This creates connections to other well-studied stable length functions in groups, namely stable commutator length and stable torsion length. As an application, we give a complete characterisation of elements whose reflection length is unbounded on powers.

Theorems & Definitions (47)

• Theorem 1
• Theorem 2: Theorem \ref{['thm cliques']}
• Proposition 3: Corollary \ref{['cor main rank one']}
• Definition 1.1
• Lemma 1.2: marquis:long
• proof
• Definition 1.3
• Definition 1.4
• Lemma 1.5: Avery2023
• Lemma 1.6