Acylindrically hyperbolic groups and their quasi-isometrically embedded subgroups

Abstract

We abstract the notion of an A/QI triple from a number of examples in geometric group theory. Such a triple (G, X, H) consists of a group G acting on a Gromov hyperbolic space X, acylindrically along a finitely generated subgroup H which is quasi-isometrically embedded by the action. Examples include strongly quasi-convex subgroups of relatively hyperbolic groups, convex cocompact subgroups of mapping class groups, many known convex cocompact subgroups of Out(Fn), and groups generated by powers of independent loxodromic WPD elements of a group acting on a Gromov hyperbolic space. We initiate the study of intersection and combination properties of A/QI triples. Under the additional hypothesis that G is finitely generated, we use a method of Sisto to show that H is stable. We apply theorems of Kapovich-Rafi and Dowdall-Taylor to analyze the Gromov boundary of an associated cone-off. We close with some examples and questions.

Some examples of separable convex‐cocompact subgroups

“…Proof. In case (1) we can form a bi-infinite quasigeodesic on which g 0 g 1 acts by translation by concatenating suitable translates of 𝑈 = [𝑦 1 , 𝑦 0 ], 𝑉 = [𝑦 0 , g 0 𝑦 0 ], and 𝑊 = [𝑦 1 , g 1 𝑦 1 ], with every other geodesic being a translate of [𝑦 0 , 𝑦 1 ]. Specifically, the ⟨g 0 g 1 ⟩-translates of the concatenation 𝑈 ⋅ 𝑉 ⋅ (g 0 Ū) ⋅ (g 0 𝑊), which joins 𝑦 1 to g 0 g 1 𝑦 1 , concatenate to form a bi-infinite path, shown in Figure 3.…”

“…Recall that for curve graphs we take 𝛿 = 100. (1). The paths concatenated to form the g 0 g 1 -axis are coloured according to their ⟨g 0 g 1 ⟩-orbits.…”

“…Then there exists a constant 𝐷 such that the following holds. Let g ∈ 𝐴 be loxodromic and let 𝑌 be its quasiaxis as in Proposition 2.1 (1). Then for any 𝑥 ∈ 𝑋 and geodesic 𝛾 = [𝑥, 𝜋 𝑌 (𝑥)] we have that 𝑁 3𝛿 (𝛾) ∩ g𝛾 has diameter at most 𝐷.…”

“…We have now reduced to the setup of [26,Lemma 4.4], from which we can conclude that 𝜎 is a cyclic shift, say 𝜎(𝑖) = 𝑖 + 𝑘 for some fixed 𝑘, and 𝑓 𝑖 ∈ 𝐸(g 𝜎(𝑖) )𝐸(g 𝜎(𝑖+1) ). Now, we have 𝐸(g 𝑗 ) = ⟨g 𝑗 ⟩ for all 𝑗 by the primitivity hypothesis and the assumption that 𝐺 is torsion free, so 𝑓 𝑖 = g 𝜎 (1) . By our hypothesis on the g 𝑖 (applied to a cyclic conjugate) we get that all 𝑚 𝑖 are 0, and therefore we get that all 𝑥 𝑖 are equal, giving us the required 𝑥.…”

Currents relative to a malnormal subgroup system

“…i"1 Ť gPFn B 8 gA i (see for instance [AM,Theorem 1.6] or [DT, Bow]). However, the double boundary B 2 p T does not contain any geodesic line whose endpoints are in distinct parabolic subgroups, which makes it a proper subspace of B 2 pF n , Aq which does not seem to be a union of cylinder sets.…”