(Non)-Morse directions in groups

Abstract

In this thesis, we study Morse directions and the Morse boundary of groups. We start by classifying the Morse boundary of all 3-manifold groups and showing that the Morse boundary of an orientable 3-manifold group only depends on the geometric decomposition of said manifold. This classification requires deep understanding of what it means to be Morse and how to manipulate Morse geodesics. While the theory of Morse boundaries and Morse geodesics is largely developed in analogy with Gromov boundaries, in the rest of this thesis, we focus on phenomena which are unique to the Morse boundary and non-hyperbolic groups. Firstly, we study the (non)-σ-compactness of the Morse boundary. The Gromov boundary of a group is always compact. In contrast, the Morse boundary of a group is only compact if it is empty or the group is hyperbolic. However, the Morse boundary exhibits more nuanced behaviour. Namely, we show that there are groups whose Morse boundary is σ-compact and groups where it is not. We give a full characterisation, which is purely combinatorial in terms of the presentation, of σ-compactness of the Morse boundary for classical small-cancellation groups. This can be used as a way to distinguish small-cancellation groups up to quasi-isometry. Furthermore, we show that for C ′ (1/9)– small-cancellation groups, the group satisfying the Morse local-to-global property is equivalent to its Morse boundary being σ-compact, implying that the topology of the Morse boundary has surprising implications on the concatenability of Morse geodesics. Secondly, in hyperbolic groups all geodesics are strongly contracting. This is precisely not the case for non-hyperbolic groups. We develop the notion of “anti-contracting geodesics segments”, that is, geodesic segments which are intrinsically not strongly contracting. We not only show that for any geodesic metric space, coning-off all anti-contracting geodesic segments results in a hyperbolic space (which we call the contraction space) but also show that if a finitely generated group acted properly on the geodesic metric space, then its action on the contraction space is non-uniformly acylindrical. Moreover, if the action was geometric and the original space was Morse-dichotomous (that is, all Morse geodesics are strongly contracting; this is the case in CAT(0) spaces and injective metric spaces), then the action on the contraction space is a universal WPD action