Complete topological descriptions of certain Morse boundaries

Abstract

We study direct limits of embedded Cantor sets and embedded \sier 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 \sier curves give homeomorphic spaces, called to $\omega$-\sier 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.

24 pages, 1 figure; added theorem that certain graph of group decompositions have $\omega$-Cantor space boundary, including fundamental groups of non-geometric graph manifolds