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 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 RELATIONS BETWEEN VARIOUS BOUNDARIES OF RELATIVELY HYPERBOLIC GROUPS

2013 ◽  
Vol 23 (07) ◽  
pp. 1551-1572 ◽  
Author(s):  
HUNG CONG TRAN
Keyword(s):  
Hyperbolic Space ◽  
Hyperbolic Groups ◽  
Hyperbolic Structure ◽  
Relatively Hyperbolic Groups ◽  
Relative Boundary ◽  
Limit Points ◽  
Relatively Hyperbolic

Suppose a group G is relatively hyperbolic with respect to a collection ℙ of its subgroups and also acts properly, cocompactly on a CAT(0) (or δ-hyperbolic) space X. The relatively hyperbolic structure provides a relative boundary ∂(G, ℙ). The CAT(0) structure provides a different boundary at infinity ∂X. In this paper, we examine the connection between these two spaces at infinity. In particular, we show that ∂(G, ℙ) is G-equivariantly homeomorphic to the space obtained from ∂X by identifying the peripheral limit points of the same type.

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 Purely exponential growth of cusp-uniform actions

2017 ◽  
Vol 39 (3) ◽  
pp. 795-831
Author(s):  
WEN-YUAN YANG
Keyword(s):  
Exponential Growth ◽  
Main Tool ◽  
Growth Function ◽  
Countable Group ◽  
Hyperbolic Groups ◽  
Growth Type ◽  
Relatively Hyperbolic Groups ◽  
Geometrically Finite ◽  
Relatively Hyperbolic ◽  
Gromov Boundary

Suppose that a countable group$G$admits a cusp-uniform action on a hyperbolic space$(X,d)$such that$G$is of divergent type. The main result of the paper is characterizing the purely exponential growth type of the orbit growth function by a condition introduced by Dal’bo, Otal and Peigné [Séries de Poincaré des groupes géométriquement finis.Israel J. Math.118(3) (2000), 109–124]. For geometrically finite Cartan–Hadamard manifolds with pinched negative curvature, this condition ensures the finiteness of Bowen–Margulis–Sullivan measures. In this case, our result recovers a theorem of Roblin (in a coarse form). Our main tool is the Patterson–Sullivan measures on the Gromov boundary of$X$, and a variant of the Sullivan shadow lemma called the partial shadow lemma. This allows us to prove that the purely exponential growth of either cones, or partial cones or horoballs is also equivalent to the Dal’bo–Otal–Peigné condition. These results are used further in a paper by the present author [W. Yang, Patterson–Sullivan measures and growth of relatively hyperbolic groups.Preprint, 2013,arXiv:1308.6326].

scholarly journals ON A GENERALIZATION OF DEHN'S ALGORITHM

2008 ◽  
Vol 18 (07) ◽  
pp. 1137-1177 ◽  
Author(s):  
OLIVER GOODMAN ◽  
MICHAEL SHAPIRO
Keyword(s):  
Finite Groups ◽  
Nilpotent Groups ◽  
Hyperbolic Groups ◽  
Fundamental Groups ◽  
Closure Properties ◽  
Relatively Hyperbolic Groups ◽  
Geometrically Finite ◽  
Geometrically Finite Groups ◽  
Relatively Hyperbolic ◽  
Infinite Subgroup

Viewing Dehn's algorithm as a rewriting system, we generalize to allow an alphabet containing letters which do not necessarily represent group elements. This extends the class of groups for which the algorithm solves the word problem to include finitely generated nilpotent groups, many relatively hyperbolic groups including geometrically finite groups and fundamental groups of certain geometrically decomposable 3-manifolds. The class has several nice closure properties. We also show that if a group has an infinite subgroup and one of exponential growth, and they commute, then it does not admit such an algorithm. We dub these Cannon's algorithms.

scholarly journals The local-to-global property for Morse quasi-geodesics

2021 ◽  
Author(s):  
Jacob Russell ◽  
Davide Spriano ◽  
Hung Cong Tran
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Type Theorem ◽  
Hyperbolic Groups ◽  
Global Property ◽  
Mapping Class ◽  
Relatively Hyperbolic Groups ◽  
Convex Cocompact ◽  
Relatively Hyperbolic ◽  
Translation Length

AbstractWe show the mapping class group, $${{\,\mathrm{CAT}\,}}(0)$$ CAT ( 0 ) groups, the fundamental groups of closed 3-manifolds, and certain relatively hyperbolic groups have a local-to-global property for Morse quasi-geodesics. This allows us to generalize combination theorems of Gitik for quasiconvex subgroups of hyperbolic groups to the stable subgroups of these groups. In the case of the mapping class group, this gives combination theorems for convex cocompact subgroups. We show a number of additional consequences of this local-to-global property, including a Cartan–Hadamard type theorem for detecting hyperbolicity locally and discreteness of translation length of conjugacy classes of Morse elements with a fixed gauge. To prove the relatively hyperbolic case, we develop a theory of deep points for local quasi-geodesics in relatively hyperbolic spaces, extending work of Hruska.

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