scholarly journals Hyperfiniteness of boundary actions of cubulated hyperbolic groups

2019 ◽  
Vol 40 (9) ◽  
pp. 2453-2466 ◽  
Author(s):  
JINGYIN HUANG ◽  
MARCIN SABOK ◽  
FORTE SHINKO
Keyword(s):  
Free Group ◽  
Hyperbolic Group ◽  
Hyperbolic Groups ◽  
Natural Action ◽  
Cube Complex ◽  
Group Case ◽  
Gromov Boundary

We show that if a hyperbolic group acts geometrically on a CAT(0) cube complex, then the induced boundary action is hyperfinite. This means that for a cubulated hyperbolic group, the natural action on its Gromov boundary is hyperfinite, which generalizes an old result of Dougherty, Jackson and Kechris for the free group case.

scholarly journals The Baum-Connes conjecture, noncommutative Poincaré duality, and the boundary of the free group

2003 ◽  
Vol 2003 (38) ◽  
pp. 2425-2445 ◽  
Author(s):  
Heath Emerson
Keyword(s):  
Free Group ◽  
Homology Class ◽  
Hyperbolic Group ◽  
Cross Product ◽  
Direct Connection ◽  
Poincaré Duality ◽  
Poincare Duality ◽  
Duality Map ◽  
The Cross ◽  
Gromov Boundary

For every hyperbolic groupΓwith Gromov boundary∂Γ, one can form the cross productC∗-algebraC(∂Γ)⋊Γ. For each such algebra, we construct a canonicalK-homology class. This class induces a Poincaré duality mapK∗(C(∂Γ)⋊Γ)→K∗+1(C(∂Γ)⋊Γ). We show that this map is an isomorphism in the case ofΓ=𝔽2, the free group on two generators. We point out a direct connection between our constructions and the Baum-Connes conjecture and eventually use the latter to deduce our result.

scholarly journals Random Subgroups of Acylindrically Hyperbolic Groups and Hyperbolic Embeddings

2017 ◽  
Vol 2019 (13) ◽  
pp. 3941-3980 ◽  
Author(s):  
Joseph Maher ◽  
Alessandro Sisto
Keyword(s):  
Normal Subgroup ◽  
Random Walks ◽  
Free Group ◽  
Hyperbolic Group ◽  
Hyperbolic Groups ◽  
Finite Collection ◽  
Small Cancellation ◽  
Cancellation Condition ◽  
Asymptotic Probability ◽  
Small Cancellation Condition

Abstract Let $G$ be an acylindrically hyperbolic group. We consider a random subgroup $H$ in $G$, generated by a finite collection of independent random walks. We show that, with asymptotic probability one, such a random subgroup $H$ of $G$ is a free group, and the semidirect product of $H$ acting on $E(G)$ is hyperbolically embedded in $G$, where $E(G)$ is the unique maximal finite normal subgroup of $G$. Furthermore, with control on the lengths of the generators, we show that $H$ satisfies a small cancellation condition with asymptotic probability one.

scholarly journals On Cannon–Thurston maps for relatively hyperbolic groups

2014 ◽  
Vol 17 (1) ◽  
Author(s):  
Yoshifumi Matsuda ◽  
Shin-ichi Oguni
Keyword(s):  
Free Group ◽  
Hyperbolic Group ◽  
Hyperbolic Groups ◽  
Relatively Hyperbolic Groups ◽  
Thurston Map ◽  
Relatively Hyperbolic Group ◽  
Relatively Hyperbolic

Abstract.Baker and Riley gave an inclusion of a free group of rank 3 in a hyperbolic group for which the Cannon–Thurston map is not well-defined. By using their result, we show that every non-elementary hyperbolic group can be included in some hyperbolic group in such a way that the Cannon–Thurston map is not well-defined. In fact we generalize their result to every non-elementary relatively hyperbolic group.

scholarly journals Cross ratios and cubulations of hyperbolic groups

2021 ◽  
Author(s):  
Jonas Beyrer ◽  
Elia Fioravanti
Keyword(s):  
Length Spectrum ◽  
Hyperbolic Groups ◽  
Optimal Length ◽  
Rigidity Result ◽  
Cube Complex ◽  
Geodesic Currents ◽  
Gromov Hyperbolic ◽  
Cube Complexes ◽  
The Relationship ◽  
Gromov Boundary

AbstractMany geometric structures associated to surface groups can be encoded in terms of invariant cross ratios on their circle at infinity; examples include points of Teichmüller space, Hitchin representations and geodesic currents. We add to this picture by studying cocompact cubulations of arbitrary Gromov hyperbolic groups G. Under weak assumptions, we show that the space of cubulations of G naturally injects into the space of G-invariant cross ratios on the Gromov boundary $$\partial _{\infty }G$$ ∂ ∞ G . A consequence of our results is that essential, hyperplane-essential, cocompact cubulations of hyperbolic groups are length-spectrum rigid, i.e. they are fully determined by their length function. This is the optimal length-spectrum rigidity result for cubulations of hyperbolic groups, as we demonstrate with some examples. In the hyperbolic setting, this constitutes a strong improvement on our previous work [4]. Along the way, we describe the relationship between the Roller boundary of a $$\mathrm{CAT(0)}$$ CAT ( 0 ) cube complex, its Gromov boundary and—in the non-hyperbolic case—the contracting boundary of Charney and Sultan. All our results hold for cube complexes with variable edge lengths.

scholarly journals ENDOMORPHISMS OF RELATIVELY HYPERBOLIC GROUPS

2008 ◽  
Vol 18 (01) ◽  
pp. 97-110 ◽  
Author(s):  
IGOR BELEGRADEK ◽  
ANDRZEJ SZCZEPAŃSKI
Keyword(s):  
Hyperbolic Group ◽  
Hyperbolic Groups ◽  
Parabolic Subgroups ◽  
Relatively Hyperbolic Groups ◽  
Property T ◽  
Relatively Hyperbolic Group ◽  
Finite Index Subgroup ◽  
Relatively Hyperbolic ◽  
Finitely Presented ◽  
Slender Group

We generalize some results of Paulin and Rips-Sela on endomorphisms of hyperbolic groups to relatively hyperbolic groups, and in particular prove the following. • If G is a nonelementary relatively hyperbolic group with slender parabolic subgroups, and either G is not co-Hopfian or Out (G) is infinite, then G splits over a slender group. • If H is a nonparabolic subgroup of a relatively hyperbolic group, and if any isometric H-action on an ℝ-tree is trivial, then H is Hopfian. • If G is a nonelementary relatively hyperbolic group whose peripheral subgroups are finitely generated, then G has a nonelementary relatively hyperbolic quotient that is Hopfian. • Any finitely presented group is isomorphic to a finite index subgroup of Out (H) for some group H with Kazhdan property (T). (This sharpens a result of Ollivier–Wise).

scholarly journals Topological flows for hyperbolic groups

2020 ◽  
pp. 1-47
Author(s):  
RYOKICHI TANAKA
Keyword(s):  
Hausdorff Dimension ◽  
Radon Measure ◽  
Hyperbolic Group ◽  
Hyperbolic Metric ◽  
Hyperbolic Groups ◽  
Group Invariant ◽  
Hyperbolic Metrics ◽  
Measure Class

Abstract Weshow that for every non-elementary hyperbolic group the Bowen–Margulis current associated with a strongly hyperbolic metric forms a unique group-invariant Radon measure class of maximal Hausdorff dimension on the boundary square. Applications include a characterization of roughly similar hyperbolic metrics via mean distortion.

scholarly journals Time complexity of the conjugacy problem in relatively hyperbolic groups

2015 ◽  
Vol 25 (05) ◽  
pp. 689-723 ◽  
Author(s):  
Inna Bumagin
Keyword(s):  
Polynomial Time ◽  
Positive Curvature ◽  
Nauk Sssr ◽  
Hyperbolic Group ◽  
Conjugacy Problem ◽  
Hyperbolic Groups ◽  
Search Problem ◽  
Relatively Hyperbolic Groups ◽  
Relatively Hyperbolic Group ◽  
Relatively Hyperbolic

If u and v are two conjugate elements of a hyperbolic group then the length of a shortest conjugating element for u and v can be bounded by a linear function of the sum of their lengths, as was proved by Lysenok in [Some algorithmic properties of hyperbolic groups, Izv. Akad. Nauk SSSR Ser. Mat. 53(4) (1989) 814–832, 912]. Bridson and Haefliger showed in [Metrics Spaces of Non-Positive Curvature (Springer-Verlag, Berlin, 1999)] that in a hyperbolic group the conjugacy problem can be solved in polynomial time. We extend these results to relatively hyperbolic groups. In particular, we show that both the conjugacy problem and the conjugacy search problem can be solved in polynomial time in a relatively hyperbolic group, whenever the corresponding problem can be solved in polynomial time in each parabolic subgroup. We also prove that if u and v are two conjugate hyperbolic elements of a relatively hyperbolic group then the length of a shortest conjugating element for u and v is linear in terms of their lengths.

scholarly journals ON PRODUCTS OF QUASICONVEX SUBGROUPS IN HYPERBOLIC GROUPS

2004 ◽  
Vol 14 (02) ◽  
pp. 173-195 ◽  
Author(s):  
ASHOT MINASYAN
Keyword(s):  
Hyperbolic Group ◽  
Hyperbolic Groups ◽  
Algebraic Representation ◽  
Quasiconvex Subgroups

An interesting question about quasiconvexity in a hyperbolic group concerns finding classes of quasiconvex subsets that are closed under finite intersections. A known example is the class of all quasiconvex subgroups [1]. However, not much is yet learned about the structure of arbitrary quasiconvex subsets. In this work we study the properties of products of quasiconvex subgroups; we show that such sets are quasiconvex and their finite intersections have a similar algebraic representation and, thus, are quasiconvex too.

A DESCRIPTION OF SOLUTIONS OF QUADRATIC EQUATIONS IN HYPERBOLIC GROUPS

1992 ◽  
Vol 02 (03) ◽  
pp. 237-274 ◽  
Author(s):  
R.I. GRIGORCHUK ◽  
I.G. LYSIONOK
Keyword(s):  
Class Group ◽  
Hyperbolic Group ◽  
Quadratic Equation ◽  
Hyperbolic Groups ◽  
Mapping Class ◽  
Parametric Solutions ◽  
Quadratic Equations ◽  
All Solutions ◽  
Finite Set ◽  
Special Mapping

A description is given for the set of solutions of a quadratic equation in a hyperbolic group. It consists of a finite set of parametric solutions of the equation which generates all solutions by the action of a group which may be interpreted as a special mapping class group of a compact surface.

scholarly journals Cannon–Thurston maps for $${{\,\mathrm{CAT}\,}}(0)$$ groups with isolated flats

2021 ◽  
Author(s):  
Beeker Benjamin ◽  
Matthew Cordes ◽  
Giles Gardam ◽  
Radhika Gupta ◽  
Emily Stark
Keyword(s):  
Automorphism Groups ◽  
Structure Theorem ◽  
Hyperbolic Group ◽  
Outer Automorphism ◽  
Hyperbolic Groups ◽  
Normal Subgroups ◽  
Infinite Index ◽  
Outer Automorphism Groups ◽  
Visual Boundaries

AbstractMahan Mitra (Mj) proved Cannon–Thurston maps exist for normal hyperbolic subgroups of a hyperbolic group (Mitra in Topology, 37(3):527–538, 1998). We prove that Cannon–Thurston maps do not exist for infinite normal hyperbolic subgroups of non-hyperbolic $${{\,\mathrm{CAT}\,}}(0)$$ CAT ( 0 ) groups with isolated flats with respect to the visual boundaries. We also show Cannon–Thurston maps do not exist for infinite infinite-index normal $${{\,\mathrm{CAT}\,}}(0)$$ CAT ( 0 ) subgroups with isolated flats in non-hyperbolic $${{\,\mathrm{CAT}\,}}(0)$$ CAT ( 0 ) groups with isolated flats. We obtain a structure theorem for the normal subgroups in these settings and show that outer automorphism groups of hyperbolic groups have no purely atoroidal $$\mathbb {Z}^2$$ Z 2 subgroups.

Sign in / Sign up

Export Citation Format

Share Document