Connectivity of Coxeter group Morse boundaries
Matthew Cordes, Ivan Levcovitz
TL;DR
The paper addresses the connectivity of Morse boundaries for Coxeter groups by introducing wide-avoidant and wide-spherical-avoidant conditions and proving that affine-free, one-ended wide-spherical-avoidant Coxeter groups have a connected and locally connected Morse boundary. It provides a complete right-angled Coxeter group (RACG) characterization and develops a Morse-geodesic criterion: a ray is Morse iff its long subpaths avoid wide subgraphs, equivalently spending uniformly bounded time in cosets of wide subgroups. Central technical tools include wall-pattern recognition (Caprace-type results), Deodhar normalizers, and a robust filter/pencil framework that builds stable connections between Morse geodesics via van-Kampen diagrams in the Davis complex. The results yield non-relatively hyperbolic examples with locally connected Morse boundaries and advance understanding of boundary behavior beyond hyperbolic groups, with potential implications for Čech cohomology and geometric group theory applications.
Abstract
We study the connectivity of Morse boundaries of Coxeter groups. We define two conditions on the defining graph of a Coxeter group: wide-avoidant and wide-spherical-avoidant. We show that wide-spherical-avoidant, one-ended, affine-free Coxeter groups have connected and locally connected Morse boundaries. On the other hand, one-ended Coxeter groups that are not wide-avoidant and not wide have disconnected Morse boundary. For the right-angled case, we get a full characterization: a one-ended right-angled Coxeter group has connected, non-empty Morse boundary if and only if it is wide-avoidant. Along the way we characterize Morse geodesic rays in affine-free Coxeter groups as those that spend uniformly bounded time in cosets of wide special subgroups.