Abstract: 

Every finitely generated group G has an associated topological space, called the Morse boundary, which captures the hyperbolic-like behavior of G at infinity. The Morse boundary was introduced by Cordes and Charney–Sultan. 

For certain affine-free Coxeter groups, the Morse boundary detects certain hierarchies in the underlying group, which we call Morse hierarchies. In this talk, we will study these hierarchies and use them to obtain a combinatorial characterization of the stabilizers of connected components of the Morse boundary, focusing on the case of right-angled Coxeter groups. This is joint work in progress with Matthew Cordes and Kim Ruane.