Complete topological descriptions of certain Morse boundaries

Abstract

We study direct limits of embedded Cantor sets and embedded Sierpiński curves. We show that under appropriate conditions on the embeddings, all limits of Cantor spaces give rise to homeomorphic spaces, called $\omega$-Cantor spaces, and, similarly, all limits of Sierpiński curves give homeomorphic spaces, called $\omega$-Sierpiński curves. We then show that the former occur naturally as Morse boundaries of right-angled Artin groups and fundamental groups of non-geometric graph manifolds, while the latter occur as Morse boundaries of fundamental groups of finite-volume, cusped hyperbolic 3-manifolds.