The local-to-global property for Morse quasi-geodesics

Abstract

We show the mapping class group, $${{\,\mathrm{CAT}\,}}(0)$$ CAT ( 0 ) groups, the fundamental groups of closed 3-manifolds, and certain relatively hyperbolic groups have a local-to-global property for Morse quasi-geodesics. This allows us to generalize combination theorems of Gitik for quasiconvex subgroups of hyperbolic groups to the stable subgroups of these groups. In the case of the mapping class group, this gives combination theorems for convex cocompact subgroups. We show a number of additional consequences of this local-to-global property, including a Cartan–Hadamard type theorem for detecting hyperbolicity locally and discreteness of translation length of conjugacy classes of Morse elements with a fixed gauge. To prove the relatively hyperbolic case, we develop a theory of deep points for local quasi-geodesics in relatively hyperbolic spaces, extending work of Hruska.

Regularity of Morse geodesics and growth of stable subgroups

“…Since ⟨𝐻 ′ , g⟩ ≅ 𝐻 ′ * ⟨g⟩, the map 𝐽 ′ → 𝐺 given by 𝑢 → 𝑢 is injective and hence 𝐽 ′ is a regular language with alphabet 𝐴 that bijects with the subgroup ⟨𝐻 ′ , g⟩. Therefore, 𝜆 𝐽 ′ = 𝜆 ⟨𝐻 ′ ,g⟩,𝐴 by Corollary 2.17, and we have [36,Corollary 3.4]). Let 𝐺 be a Morse local-to-global group.…”

“…By Theorem 6.5, for every open neighborhood 𝑈 of 𝑥 ∈ 𝜕 * 𝐺, there exists some Morse element g ∈ 𝐺, such that g + ∈ 𝑈. By [36,Corollary 3.6], there is ℎ ∈ 𝐻 such that ℎ is a Morse element of 𝐺. Using Lemma 6.3, there exists some 𝑚 ∈ ℕ such that g 𝑚 ℎ + ∈ 𝑈.…”

“…Morse local-to-global spaces are therefore spaces where this local-to-global property for Morse quasi-geodesics does hold. Definition 2.20 [36,Definition 2.12]. A metric space 𝑋 is a Morse local-to-global space if for every Morse gauge 𝑀 and constants 𝑘 ⩾ 1, 𝑐 ⩾ 0, there exists a local scale 𝐵 ⩾ 0, Morse gauge 𝑀 ′ , and constants 𝑘 ′ ⩾ 1, 𝑐 ′ ⩾ 0 so that every (𝐵; 𝑀; 𝑘, 𝑐)-local Morse quasi-geodesic is a global (𝑀 ′ ; 𝑘 ′ , 𝑐 ′ )-Morse quasi-geodesic.…”

“…Gromov showed that hyperbolic spaces are characterized by local quasi‐geodesics being global quasi‐geodesics [25]; the Morse local‐to‐global property is a generalization of this phenomena. Morse local‐to‐global groups were introduced by Russell, Spriano, and Tran who showed that they include the mapping class group of an orientable, finite‐type surface, all cocompact $$\operatorname{CAT}(0)$$ groups, the fundamental group of any closed 3‐manifold, and any group hyperbolic relative to Morse local‐to‐global groups [36]. We will describe below the two places in the proof of Theorem A where the Morse local‐to‐global property is used.…”

“…Given an infinite‐index stable subgroup $$H$$, we use either the virtual torsion freeness of $$G$$ or the residually finiteness of $$H$$ to produce an element $$g \in G$$ and a finite‐index subgroup $$H^{\prime } < H$$ so that $$\langle H^{\prime },g\rangle \cong H^{\prime } \ast \langle g \rangle$$. In both cases, this relies on combination theorems for Morse local‐to‐global groups proved by Russell, Spriano, and Tran [36]. Since finite‐index subgroups of stable subgroups are stable, we can apply Theorem E to $$H^{\prime }$$ to produce a regular language $$J_{H^{\prime }}$$ of geodesic words that uniquely represent the elements of $$H^{\prime }$$.…”