Connectivity of Coxeter group Morse boundaries

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.