Hyperbolic Groups Research Articles

Frattini subgroups of hyperbolic-like groups

We study Frattini subgroups of various generalizations of hyperbolic groups. For any countable group G admitting a general type action on a hyperbolic space S, we show that the induced action of the Frattini subgroup Φ(G) on S has bounded orbits. This implies that Φ(G) is “small” compared to G; in particular, |G:Φ(G)|=∞. In contrast, for any finitely generated non-cyclic group Q with Φ(Q)={1}, we construct an infinite lacunary hyperbolic group L such that L/Φ(L)≅Q; in particular, the Frattini subgroup of an infinite lacunary hyperbolic group can have finite index. As an application, we obtain the first examples of invariably generated, infinite, lacunary hyperbolic groups.

Contracting boundary of a cusped space

Let [Formula: see text] be a finitely generated group. Cashen and Mackay proved that if the contracting boundary of [Formula: see text] with the topology of fellow traveling quasi-geodesics is compact then [Formula: see text] is a hyperbolic group. Let [Formula: see text] be a finite collection of finitely generated infinite index subgroups of [Formula: see text]. Let [Formula: see text] be the cusped space obtained by attaching combinatorial horoballs to each left coset of elements of [Formula: see text]. In this article, we prove that if the combinatorial horoballs are contracting and [Formula: see text] has compact contracting boundary then [Formula: see text] is hyperbolic relative to [Formula: see text].

Simple 𝑝-adic Lie groups with abelian Lie algebras

Abstract For each prime 𝑝 and each positive integer 𝑑, we construct the first examples of second countable, topologically simple 𝑝-adic Lie groups of dimension 𝑑 whose Lie algebras are abelian. This answers several questions of Glöckner and Caprace–Monod. The proof relies on a generalization of small cancellation methods that applies to central extensions of acylindrically hyperbolic groups.

Local connectedness of boundaries for relatively hyperbolic groups

AbstractLet be a relatively hyperbolic group pair that is relatively one ended. Then, the Bowditch boundary of is locally connected. Bowditch previously established this conclusion under the additional assumption that all peripheral subgroups are finitely presented, either one or two ended, and contain no infinite torsion subgroups. We remove these restrictions; we make no restriction on the cardinality of and no restriction on the peripheral subgroups .

On isomorphisms to a free group and beyond

The isomorphism problem for infinite finitely presented groups is probably the hardest among standard algorithmic problems in group theory. Classes of groups where it has been completely solved are nilpotent groups, hyperbolic groups, and limit groups. In this short paper, we address the problem of isomorphism to particular groups, including free groups. We also address the algorithmic problem of embedding a finitely presented group in a given limit group.

Action rigidity for free products of hyperbolic manifold groups

Action rigidity for free products of hyperbolic manifold groups

Hyperbolic one-relator groups

Abstract We introduce two families of two-generator one-relator groups called primitive extension groups and show that a one-relator group is hyperbolic if its primitive extension subgroups are hyperbolic. This reduces the problem of characterizing hyperbolic one-relator groups to characterizing hyperbolic primitive extension groups. These new groups, moreover, admit explicit decompositions as graphs of free groups with adjoined roots. In order to obtain this result, we characterize $2$ -free one-relator groups with exceptional intersection in terms of Christoffel words, show that hyperbolic one-relator groups have quasi-convex Magnus subgroup, and build upon the one-relator tower machinery developed in previous work of the author.

Special cubulation of strict hyperbolization

We prove that the Gromov hyperbolic groups obtained by the strict hyperbolization procedure of Charney and Davis are virtually compact special, hence linear and residually finite. Our strategy consists in constructing an action of a hyperbolized group on a certain dual CAT(0)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\operatorname {CAT}(0)$\\end{document} cubical complex. As a result, all the common applications of strict hyperbolization are shown to provide manifolds with virtually compact special fundamental group. In particular, we obtain examples of closed negatively curved Riemannian manifolds whose fundamental groups are linear and virtually algebraically fiber.

Hyperbolicity and bounded-valued cohomology

We generalise a theorem of Gersten on surjectivity of the restriction map in ℓ∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell ^{\\infty }$$\\end{document}-cohomology of groups. This leads to applications on subgroups of hyperbolic groups, quasi-isometric distinction of finitely generated groups and ℓ∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell ^{\\infty }$$\\end{document}-cohomology calculations for some well-known classes of groups. Along the way, we obtain hyperbolicity criteria for groups of type FP2(Q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$FP_2({{\\mathbb {Q}}})$$\\end{document} and for those satisfying a rational homological linear isoperimetric inequality, answering a question of Arora and Martínez-Pedroza.

Equidistribution of hyperbolic groups in homogeneous spaces

We prove that infinite orbits of Zariski dense hyperbolic groups equidistribute in homogeneous spaces, in the sense that the family of measures obtained by averaging along spheres in the Cayley graph converges to Haar measure.

Commensurating HNN-extensions: Hierarchical hyperbolicity and biautomaticity

We construct a \mathrm{CAT}(0) hierarchically hyperbolic group (HHG) acting geometrically on the product of a hyperbolic plane and a locally-finite tree which is not biautomatic. This gives the first example of an HHG which is not biautomatic, the first example of a non-biautomatic \mathrm{CAT}(0) group of flat-rank 2 , and the first example of an HHG which is injective but not Helly. Our proofs heavily utilise the space of geodesic currents for a hyperbolic surface.

Towers and the First-order Theories of Hyperbolic Groups

This paper is devoted to the first-order theories of torsion-free hyperbolic groups. One of its purposes is to review some results and to provide precise and correct statements and definitions, as well as some proofs and new results. A key concept is that of a tower (Sela) or NTQ system (Kharlampovich-Myasnikov). We discuss them thoroughly. We state and prove a new general theorem which unifies several results in the literature: elementarily equivalent torsion-free hyperbolic groups have isomorphic cores (Sela); if H H is elementarily embedded in a torsion-free hyperbolic group G G , then G G is a tower over H H relative to H H (Perin); free groups (Perin-Sklinos, Ould-Houcine), and more generally free products of prototypes and free groups, are homogeneous. The converse to Sela and Perin’s results just mentioned is true. This follows from the solution to Tarski’s problem on elementary equivalence of free groups, due independently to Sela and Kharlampovich-Myasnikov, which we treat as a black box throughout the paper. We present many examples and counterexamples, and we prove some new model-theoretic results. We characterize prime models among torsion-free hyperbolic groups, and minimal models among elementarily free groups. Using Fraïssé’s method, we associate to every torsion-free hyperbolic group H H a unique homogeneous countable group M {\mathcal {M}} in which any hyperbolic group H ′ H’ elementarily equivalent to H H has an elementary embedding. In an appendix we give a complete proof of the fact, due to Sela, that towers over a torsion-free hyperbolic group H H are H H -limit groups.

Acylindrically Hyperbolic Groups and Their Quasi-Isometrically Embedded Subgroups

Acylindrically Hyperbolic Groups and Their Quasi-Isometrically Embedded Subgroups

The compressed conjugacy problem in relatively hyperbolic groups

We prove that the compressed conjugacy problem in a group that is hyperbolic relative to a collection of free abelian subgroups is solvable in polynomial time.

Mirror stabilizers for lattice complex hyperbolic triangle groups

Mirror stabilizers for lattice complex hyperbolic triangle groups

Structure invariant properties of the hierarchically hyperbolic boundary

We prove several topological and dynamical properties of the boundary of a hierarchically hyperbolic group are independent of the specific hierarchically hyperbolic structure. This is accomplished by proving that the boundary is invariant under a “maximization” procedure introduced by the first two authors and Durham.

Computable Scott sentences and the weak Whitehead problem for finitely presented groups

We prove that if A is a computable Hopfian finitely presented structure, then A has a computable d-Σ2 Scott sentence if and only if the weak Whitehead problem for A is decidable. We use this to infer that every hyperbolic group as well as any polycyclic-by-finite group has a computable d-Σ2 Scott sentence, thus covering two main classes of finitely presented groups. Our proof also implies that every weakly Hopfian finitely presented group is strongly defined by its ∃+-types, a question which arose in a different context.

A non-Hopfian relatively hyperbolic group with respect to a Hopfian subgroup

We produce an example demonstrating that every finitely generated relatively hyperbolic group with respect to a collection of Hopfian subgroups need not be Hopfian. This answers a question of Osin (2006) in the negative.

On the Čech cohomology of Morse boundaries

We consider cusped hyperbolic n–manifolds, and compute Čech cohomology groups of the Morse boundaries of their fundamental groups. In particular, we show that the reduced Čech cohomology with real coefficients vanishes in dimension at most n−3 and does not vanish in dimension n−2. A similar result holds for relatively hyperbolic groups with virtually nilpotent peripherals and Bowditch boundary homeomorphic to a sphere; these include all non-uniform lattices in rank–1 simple Lie groups.

Graphs of curves and arcs quasi-isometric to big mapping class groups

Following the works of Rosendal and Mann and Rafi, we try to answer the following question: when is the mapping class group of an infinite-type surface quasi-isometric to a graph whose vertices are curves on that surface? With the assumption of tameness as defined by Mann and Rafi, we describe a necessary and sufficient condition, called translatability, for a geometrically non-trivial big mapping class group to admit such a quasi-isometry. In addition, we show that the mapping class group of the plane minus a Cantor set is quasi-isometric to the loop graph defined by Bavard, which we believe represents the first known example of a hyperbolic mapping class group that is not virtually free.