Let 𝐺 be a finitely generated group. We show that for any finite symmetric generating set 𝐴, the language consisting of all geodesics in Cay(𝐺, 𝐴) with the contracting property is a regular language. An immediate consequence is that the existence of an infinite contracting geodesic in a Cayley graph of a finitely generated group implies the existence of a contracting element. In particular, torsion groups cannot contain an infinite contracting geodesic. As an application, this implies that any finitely generated group containing an infinite contracting geodesic must be either virtually ℤ or acylindrically hyperbolic.

Given a sublinear function κ, the κ-Morse boundary ∂κG of a CAT(0) group was introduced by Qing and Rafi and shown to be a quasiisometry invariant and a metrizable space. In this paper, we prove several properties of the κ-Morse boundary that further generalize useful properties of the Gromov boundary. We first show that if X is a proper CAT(0) space, then ∂κX is a strong visibility space. We also show that any CAT(0) group G with nonempty κ-boundary contains a rank one isometry; furthermore, the subspace consisting of strongly contracting rays is a dense subspace of ∂κG. These results generate several applications: any CAT(0) group with nonempty κ-Morse boundary is an acylindrically hyperbolic group; the collection of rank one isometries G contains is exponentially generic. Furthermore, we show that a homeomorphism f : ∂κG → ∂κG comes from a quasi-isometry if and only if f is Morse quasi-möbius and stable. Lastly, we characterize exactly when the sublinearly Morse boundary is a compact space.

Let G be a finitely generated group. We show that for any generating set A, the language consisting of all geodesics in Cay(G, A) with a contracting property is a regular language. Also, for a group G acting properly and cocompactly on a CAT(0) space X, we show that for any generating set A, the language consisting of all geodesics in Cay(G, A) with a D-contracting quasi geodesic image in X is a regular language.

We prove that the sublinearly Morse boundary of every known cubulated group continuously injects in the Gromov boundary of a certain hyperbolic graph. We also show that for all CAT(0) cube complexes, convergence to sublinearly Morse geodesic rays has a simple combinatorial description using the hyperplanes crossed by such sequences. As an application of this combinatorial description, we show that a certain subspace of the Roller boundary continously surjects on the subspace of the visual boundary consisting of sublinearly Morse geodesic rays.