Regular languages for contracting geodesics

Abstract

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.

Regularity of Morse geodesics and growth of stable subgroups

“…Theorem D provides an extension of a result by Eike and Zalloum who showed that strongly contracting geodesics with a fixed parameter form a regular language in any finitely generated group [17]. Strongly contracting geodesics are a stronger form of hyperbolic-like behavior in a group compared to Morse geodesics.…”

“…We first prove 𝑢𝑞 = 𝑢𝑤𝑎 is a geodesic word in Cay(𝐺, 𝐴). This part of the proof closely follows [17] and the cone types section of [7]. Recall, for 𝜔 ∈ 𝐴 ⋆ , 𝓁(𝜔) denotes the word length in the free monoid 𝐴 ⋆ , while |𝜔| denotes the length of a geodesic word in Cay(𝐺, 𝐴) that represents the group element 𝜔.…”

“…We first prove $$uq = uwa$$ is a geodesic word in $$\operatorname{Cay}(G,A)$$. This part of the proof closely follows [17] and the cone types section of [7]. Recall, for $$\omega \in A^\star$$, $$\ell (\omega )$$ denotes the word length in the free monoid $$A^\star$$, while $$| \overline{\omega }|$$ denotes the length of a geodesic word in $$\operatorname{Cay}(G,A)$$ that represents the group element $$\overline{\omega }$$.…”