Fixed-point-free involutions on the boundaries of Right-angled Coxeter Groups

TL;DR

The paper addresses a variation of Bogdan Nica's conjecture by proving that right-angled Coxeter groups (RACGs) admit a fixed-point-free involution on the boundary of their associated CAT(0) Davis complex. The authors construct an order-two element $\gamma=\prod_{a\in S} a$ from a maximal spherical subset $S\subset V$ and show that $\gamma$ fixes only a single point $x_0$ in $\Sigma$, with the local action equivalent to the antipodal map on $\mathbb{R}^{S}$. Consequently, the induced boundary action on $\partial\Gamma$ has no fixed points, yielding a fixed-point-free involution of the boundary. This provides a targeted result for RACGs and offers techniques that might illuminate the original fixed-point-free boundary problem for word-hyperbolic groups. The approach connects Coxeter group theory, CAT(0) geometry, and boundary dynamics, with potential implications for EZ-structures and boundary rigidity.

Abstract

About 20 years ago, Bogdan Nica conjectured that the boundary of any word-hyperbolic group admits admits a fixed-point-free involution. In this very short article, we prove a variation of the conjecture, replacing word-hyperbolic groups with Right-angled coxeter groups. This is not a solution to Nica's conjecture, but hopefully the techniques will shed some light regarding possible approaches to the problem.

Theorems & Definitions (5)

• Theorem 1: davis_geometry_2008, Theorem 12.2.1 (credited to Gromov)
• Proposition 1
• proof : Proof of Proposition.
• Theorem 2
• proof