Wide-avoidant characterization of connected Morse boundaries for Coxeter groups

Establish that a Coxeter group has connected, non-empty Morse boundary if and only if it is one-ended and wide-avoidant, where wide-avoidant means that for every wide induced subgraph Δ of the Coxeter graph Γ and every pair of vertices s,t in Γ, there exists a path from s to t whose intersection with Δ is contained in {s,t}.

Background

The paper introduces the wide-avoidant condition on Coxeter graphs and proves several partial results showing when Morse boundaries are connected or disconnected. In particular, it proves the full characterization for right-angled Coxeter groups and gives sufficient conditions for general Coxeter groups under an affine-free hypothesis.

The conjecture posits that wide-avoidance together with one-endedness exactly characterizes when a Coxeter group’s Morse boundary is connected and non-empty, thereby aiming for a clean, complete criterion beyond the classes established in the paper.