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Register a new userImmutability is not uniformly decidable in hyperbolic groups
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A finitely generated subgroup H of a torsion-free hyperbolic group G is called immutable if there are only finitely many conjugacy classes of injections of H into G. We show that there is no uniform algorithm to recognize immutability, answering a uniform version of a question asked by the authors.
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Sela proved every torsion-free one-ended hyperbolic group is coHopfian. We prove that there exist torsion-free one-ended hyperbolic groups that are not commensurably coHopfian. In particular, we show that the fundamental group of every simple surface amalgam is not commensurably coHopfian.
We examine residual properties of word-hyperbolic groups, adapting a method introduced by Darren Long to study the residual properties of Kleinian groups.
For every group $G$, we introduce the set of hyperbolic structures on $G$, denoted $mathcal{H}(G)$, which consists of equivalence classes of (possibly infinite) generating sets of $G$ such that the corresponding Cayley graph is hyperbolic; two generating sets of $G$ are equivalent if the corresponding word metrics on $G$ are bi-Lipschitz equivalent. Alternatively, one can define hyperbolic structures in terms of cobounded $G$-actions on hyperbolic spaces. We are especially interested in the subset $mathcal{AH}(G)subseteq mathcal{H}(G)$ of acylindrically hyperbolic structures on $G$, i.e., hyperbolic structures corresponding to acylindrical actions. Elements of $mathcal{H}(G)$ can be ordered in a natural way according to the amount of information they provide about the group $G$. The main goal of this paper is to initiate the study of the posets $mathcal{H}(G)$ and $mathcal{AH}(G)$ for various groups $G$. We discuss basic properties of these posets such as cardinality and existence of extremal elements, obtain several results about hyperbolic structures induced from hyperbolically embedded subgroups of $G$, and study to what extent a hyperbolic structure is determined by the set of loxodromic elements and their translation lengths.
We generalize a version of small cancellation theory to the class of acylindrically hyperbolic groups. This class contains many groups which admit some natural action on a hyperbolic space, including non-elementary hyperbolic and relatively hyperbolic groups, mapping class groups, and groups of outer automorphisms of free groups. Several applications of this small cancellation theory are given, including to Frattini subgroups and Kazhdan constants, the construction of various exotic quotients, and to approximating acylindrically hyperbolic groups in the topology of marked group presentations.
Let $Gamma$ be a torsion-free hyperbolic group. We study $Gamma$--limit groups which, unlike the fundamental case in which $Gamma$ is free, may not be finitely presentable or geometrically tractable. We define model $Gamma$--limit groups, which always have good geometric properties (in particular, they are always relatively hyperbolic). Given a strict resolution of an arbitrary $Gamma$--limit group $L$, we canonically construct a strict resolution of a model $Gamma$--limit group, which encodes all homomorphisms $Lto Gamma$ that factor through the given resolution. We propose this as the correct framework in which to study $Gamma$--limit groups algorithmically. We enumerate all $Gamma$--limit groups in this framework.
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