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 !-Cantor spaces, and, similarly, all limits of Sierpiński curves give homeomorphic spaces, called !-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.

Morse boundaries of graphs of groups with finite edge groups

“…We show that, in contrast, the Morse boundaries of C ′ (1/6)-small-cancellation groups exhibit a variety of behaviours. As suggested in [CCS19], we show that there indeed exist C ′ (1/6)-smallcancellation groups with non-σ-compact Morse boundary. Further we show that not all infinitely presented C ′ (1/6)-groups have non-σ-compact Morse boundary.…”

“…The Morse boundary is neither compact nor metrizable for non-hyperbolic groups [CD19,Mur19], but for all known examples so far the Morse boundary is σ-compact, such as in the cases where it has been fully described [CCS19,Zbi22], and in all groups (quasi-isometric to a space) where all Morse rays are strongly contracting such as CAT(0) groups and coarsely helly groups, including hierarchically hyperbolic groups [Sul14,HHP20,SZ22].…”

On the Čech cohomology of Morse boundaries

“…Our main theorem is in a similar line of research as the papers [CCS19,Zbi21,Zbi22], where Morse boundaries of certain groups, including 3-manifold groups, are fully described. A similar goal seems out of reach in our case (except when n = 3, as shown in [CCS19]), but we also think of our theorem as a proof of concept: even when the Morse boundary is not "fully describable", powerful topological invariants can sometimes still be computed.…”