Wide-avoidant connectedness characterization for Morse boundaries of Coxeter groups

Determine whether every Coxeter group has connected, non-empty Morse boundary if and only if it is one-ended and wide-avoidant. Concretely, establish the bidirectional implication between connected, non-empty Morse boundary and the wide-avoidant property for all Coxeter groups, thereby providing a complete characterization of when the Morse boundary is connected.

Background

The paper introduces the notions of wide and wide-avoidant Coxeter groups and proves several partial results toward understanding the connectivity of Morse boundaries. In particular, it shows that one-ended, affine-free, wide-spherical-avoidant Coxeter groups have connected and locally connected Morse boundaries, and fully characterizes the right-angled case: a one-ended right-angled Coxeter group has connected, non-empty Morse boundary if and only if it is wide-avoidant.

Motivated by these results and the partial characterizations, the authors propose a complete characterization in general, asserting that connected, non-empty Morse boundary should coincide exactly with the one-ended wide-avoidant condition for all Coxeter groups.