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We introduce an obstruction to the existence of a coarse embedding of a given group or space into a hyperbolic group, or more generally into a hyperbolic graph of bounded degree. The condition we consider is admitting exponentially many fat bigons, and it is preserved by a coarse embedding between graphs with bounded degree. Groups with exponential growth and linear divergence (such as direct products of two groups one of which has exponential growth, solvable groups that are not virtually nilpotent, and uniform higher-rank lattices) have this property and hyperbolic graphs do not, so the former cannot be coarsely embedded into the latter. Other examples include certain lacunary hyperbolic and certain small cancellation groups.
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We characterize the Lie groups with finitely many connected components that are $O(u)$-bilipschitz equivalent (almost quasiisometric in the sense that the sublinear function $u$ replaces the additive bounds of quasiisometry) to the real hyperbolic space, or to the complex hyperbolic plane. The characterizations are expressed in terms of deformations of Lie algebras and in terms of pinching of sectional curvature of left-invariant Riemannian metrics in the real case. We also compare sublinear bilipschitz equivalence and coarse equivalence, and prove that every coarse equivalence between the logarithmic coarse structures of geodesic spaces is a $O(log)$-bilipschitz equivalence. The Lie groups characterized are exactly those whose logarithmic coarse structure is equivalent to that of a real hyperbolic space or the complex hyperbolic plane.
We present a contribution to the structure theory of locally compact groups. The emphasis is on compactly generated locally compact groups which admit no infinite discrete quotient. It is shown that such a group possesses a characteristic cocompact subgroup which is either connected or admits a non-compact non-discrete topologically simple quotient. We also provide a description of characteristically simple groups and of groups all of whose proper quotients are compact. We show that Noetherian locally compact groups without infinite discrete quotient admit a subnormal series with all subquotients compact, compactly generated Abelian, or compactly generated topologically simple. Two appendices introduce results and examples around the concept of quasi-product.
We build quasi--isometry invariants of relatively hyperbolic groups which detect the hyperbolic parts of the group; these are variations of the stable dimension constructions previously introduced by the authors. We prove that, given any finite collection of finitely generated groups $mathcal{H}$ each of which either has finite stable dimension or is non-relatively hyperbolic, there exist infinitely many quasi--isometry types of one--ended groups which are hyperbolic relative to $mathcal{H}$. The groups are constructed using small cancellation theory over free products.
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.
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.
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