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 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 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 Asymptotically isometric metrics on relatively hyperbolic groups and marked length spectrum

2015 ◽  
Vol 07 (02) ◽  
pp. 345-359 ◽  
Author(s):  
Koji Fujiwara
Keyword(s):  
Hyperbolic Group ◽  
Length Spectrum ◽  
Hyperbolic Groups ◽  
Fundamental Groups ◽  
Relatively Hyperbolic Groups ◽  
Universal Covers ◽  
Relatively Hyperbolic Group ◽  
Marked Length Spectrum ◽  
Isospectral Problem ◽  
Relatively Hyperbolic

We prove asymptotically isometric, coarsely geodesic metrics on a toral relatively hyperbolic group are coarsely equal. The theorem applies to all lattices in SO (n, 1). This partly verifies a conjecture by Margulis. In the case of hyperbolic groups/spaces, our result generalizes a theorem by Furman and a theorem by Krat. We discuss an application to the isospectral problem for the length spectrum of Riemannian manifolds. The positive answer to this problem has been known for several cases. Most of them have hyperbolic fundamental groups. We do not solve the isospectral problem in the original sense, but prove the universal covers are (1, C)-quasi-isometric if the fundamental group is a toral relatively hyperbolic group.

scholarly journals RELATIVELY HYPERBOLIC GROUPS

2012 ◽  
Vol 22 (03) ◽  
pp. 1250016 ◽  
Author(s):  
B. H. BOWDITCH
Keyword(s):  
Hyperbolic Space ◽  
Limit Point ◽  
Hyperbolic Group ◽  
Hyperbolic Groups ◽  
Relatively Hyperbolic Groups ◽  
Geometrically Finite ◽  
Hyperbolic Graph ◽  
Finite Action ◽  
Relatively Hyperbolic Group ◽  
Relatively Hyperbolic

In this paper we develop some of the foundations of the theory of relatively hyperbolic groups as originally formulated by Gromov. We prove the equivalence of two definitions of this notion. One is essentially that of a group admitting a properly discontinuous geometrically finite action on a proper hyperbolic space, that is, such that every limit point is either a conical limit point or a bounded parabolic point. The other is that of a group which admits a cofinite action on a connected fine hyperbolic graph. We define a graph to be "fine" if there are only finitely many circuits a given length containing any given edge, and we develop some of the properties of this notion. We show how a relatively hyperbolic group can be assumed to act on a proper hyperbolic space of a particular geometric form. We define the boundary of a relatively hyperbolic group, and show that the limit set of any geometrically finite action of the group is equivariantly homeomorphic to this boundary. This generalizes a result of Tukia for geometrically finite kleinian groups. We also describe when the boundary is connected.

scholarly journals PARABOLIC GROUPS ACTING ON ONE-DIMENSIONAL COMPACT SPACES

2005 ◽  
Vol 15 (05n06) ◽  
pp. 893-906
Author(s):  
FRANÇOIS DAHMANI
Keyword(s):  
Hyperbolic Group ◽  
Surface Groups ◽  
Parabolic Subgroups ◽  
Maximal Parabolic Subgroup ◽  
One Dimensional ◽  
Compact Spaces ◽  
Dimensional Boundary ◽  
Relatively Hyperbolic Group ◽  
Relatively Hyperbolic ◽  
Maximal Parabolic Subgroups

Given a class of compact spaces, we ask which groups can be maximal parabolic subgroups of a relatively hyperbolic group whose boundary is in the class. We investigate the class of one-dimensional connected boundaries. We get that any non-torsion infinite finitely-generated group is a maximal parabolic subgroup of some relatively hyperbolic group with connected one-dimensional boundary without global cut point. For boundaries homeomorphic to a Sierpinski carpet or a 2-sphere, the only maximal parabolic subgroups allowed are virtual surface groups (hyperbolic, or virtually ℤ + ℤ).

scholarly journals SLP compression for solutions of equations with constraints in free and hyperbolic groups

2015 ◽  
Vol 25 (01n02) ◽  
pp. 81-111
Author(s):  
Volker Diekert ◽  
Olga Kharlampovich ◽  
Atefeh Mohajeri Moghaddam
Keyword(s):  
Free Groups ◽  
Hyperbolic Group ◽  
Hyperbolic Groups ◽  
Boolean Formula ◽  
System Of Equations ◽  
Exponential Bound ◽  
Word Equation ◽  
Relatively Hyperbolic Group ◽  
Solutions Of Equations ◽  
Relatively Hyperbolic

The paper is a part of an ongoing program which aims to show that the problem of satisfiability of a system of equations in a free group (hyperbolic or even toral relatively hyperbolic group) is NP-complete. For that, we study compression of solutions with straight-line programs (SLPs) as suggested originally by Plandowski and Rytter in the context of a single word equation. We review some basic results on SLPs and give full proofs in order to keep this fundamental part of the program self-contained. Next we study systems of equations with constraints in free groups and more generally in free products of abelian groups. We show how to compress minimal solutions with extended Parikh-constraints. This type of constraints allows to express semi-linear conditions as e.g. alphabetic information. The result relies on some combinatorial analysis and has not been shown elsewhere. We show similar compression results for Boolean formula of equations over a torsion-free δ-hyperbolic group. The situation is much more delicate than in free groups. As byproduct we improve the estimation of the "capacity" constant used by Rips and Sela in their paper "Canonical representatives and equations in hyperbolic groups" from a double-exponential bound in δ to some single-exponential bound. The final section shows compression results for toral relatively hyperbolic groups using the work of Dahmani: We show that given a system of equations over a fixed toral relatively hyperbolic group, for every solution of length N there is an SLP for another solution such that the size of the SLP is bounded by some polynomial p(s + log N) where s is the size of the system.

scholarly journals Towers and elementary embeddings in toral relatively hyperbolic groups

2021 ◽  
pp. 1-27
Author(s):  
Christopher Perez
Keyword(s):  
Free Groups ◽  
Hyperbolic Group ◽  
Hyperbolic Groups ◽  
Geometric Structures ◽  
Relatively Hyperbolic Groups ◽  
First Order ◽  
Elementary Embeddings ◽  
Elementary Submodel ◽  
Relatively Hyperbolic ◽  
Torsion Free

In a remarkable series of papers, Zlil Sela classified the first-order theories of free groups and torsion-free hyperbolic groups using geometric structures he called towers. It was later proved by Chloé Perin that if [Formula: see text] is an elementarily embedded subgroup (or elementary submodel) of a torsion-free hyperbolic group [Formula: see text], then [Formula: see text] is a tower over [Formula: see text]. We prove a generalization of Perin’s result to toral relatively hyperbolic groups using JSJ and shortening techniques.

scholarly journals DISMANTLABLE CLASSIFYING SPACE FOR THE FAMILY OF PARABOLIC SUBGROUPS OF A RELATIVELY HYPERBOLIC GROUP

2017 ◽  
Vol 18 (2) ◽  
pp. 329-345 ◽  
Author(s):  
Eduardo Martínez-Pedroza ◽  
Piotr Przytycki
Keyword(s):  
Hyperbolic Group ◽  
Finite Collection ◽  
Parabolic Subgroups ◽  
Classifying Space ◽  
New Approach ◽  
The Family ◽  
Free Case ◽  
Finite Subgroups ◽  
Relatively Hyperbolic Group ◽  
Relatively Hyperbolic

Let $G$ be a group hyperbolic relative to a finite collection of subgroups ${\mathcal{P}}$. Let ${\mathcal{F}}$ be the family of subgroups consisting of all the conjugates of subgroups in ${\mathcal{P}}$, all their subgroups, and all finite subgroups. Then there is a cocompact model for $E_{{\mathcal{F}}}G$. This result was known in the torsion-free case. In the presence of torsion, a new approach was necessary. Our method is to exploit the notion of dismantlability. A number of sample applications are discussed.

scholarly journals From local to global conjugacy of subgroups of relatively hyperbolic groups

2017 ◽  
Vol 27 (03) ◽  
pp. 299-314
Author(s):  
Oleg Bogopolski ◽  
Kai-Uwe Bux
Keyword(s):  
Finite Index ◽  
Minimal Length ◽  
Hyperbolic Groups ◽  
Index Subgroup ◽  
Finitely Generated ◽  
Limit Group ◽  
Relatively Hyperbolic Groups ◽  
Finitely Generated Group ◽  
Finite Index Subgroup ◽  
Relatively Hyperbolic

Suppose that a finitely generated group [Formula: see text] is hyperbolic relative to a collection of subgroups [Formula: see text]. Let [Formula: see text] be subgroups of [Formula: see text] such that [Formula: see text] is relatively quasiconvex with respect to [Formula: see text] and [Formula: see text] is not parabolic. Suppose that [Formula: see text] is elementwise conjugate into [Formula: see text]. Then there exists a finite index subgroup of [Formula: see text] which is conjugate into [Formula: see text]. The minimal length of the conjugator can be estimated. In the case, where [Formula: see text] is a limit group, it is sufficient to assume only that [Formula: see text] is a finitely generated and [Formula: see text] is an arbitrary subgroup of [Formula: see text].

scholarly journals Subgroups of relatively hyperbolic groups of Bredon cohomological dimension 2

2017 ◽  
Vol 20 (6) ◽  
Author(s):  
Eduardo Martínez-Pedroza
Keyword(s):  
Cohomological Dimension ◽  
Hyperbolic Groups ◽  
Relatively Hyperbolic Groups ◽  
Remarkable Result ◽  
Dimension 2 ◽  
Relatively Hyperbolic ◽  
Finitely Presented

AbstractA remarkable result of Gersten states that the class of hyperbolic groups of cohomological dimension 2 is closed under taking finitely presented (or more generally

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