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 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 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.

scholarly journals Splittings and automorphisms of relatively hyperbolic groups

2015 ◽  
Vol 9 (2) ◽  
pp. 599-663 ◽  
Author(s):  
Vincent Guirardel ◽  
Gilbert Levitt
Keyword(s):  
Hyperbolic Groups ◽  
Relatively Hyperbolic Groups ◽  
Relatively Hyperbolic

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 QUASI-ISOMETRIES, BOUNDARIES AND JSJ-DECOMPOSITIONS OF RELATIVELY HYPERBOLIC GROUPS

2013 ◽  
Vol 05 (04) ◽  
pp. 451-475 ◽  
Author(s):  
BRADLEY W. GROFF
Keyword(s):  
Hyperbolic Groups ◽  
Isometry Groups ◽  
Geometric Structures ◽  
Relatively Hyperbolic Groups ◽  
Jsj Decomposition ◽  
Outer Automorphisms ◽  
Jsj Decompositions ◽  
Relatively Hyperbolic

We demonstrate the quasi-isometry invariance of two important geometric structures for relatively hyperbolic groups: the coned space and the cusped space. As applications, we produce a JSJ-decomposition for relatively hyperbolic groups which is invariant under quasi-isometries and outer automorphisms, as well as a related splitting of the quasi-isometry groups of relatively hyperbolic groups.

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 THE COARSE BAUM–CONNES CONJECTURE FOR RELATIVELY HYPERBOLIC GROUPS

2012 ◽  
Vol 04 (01) ◽  
pp. 99-113 ◽  
Author(s):  
TOMOHIRO FUKAYA ◽  
SHIN-ICHI OGUNI
Keyword(s):  
Finite Family ◽  
Hyperbolic Groups ◽  
Relatively Hyperbolic Groups ◽  
Proper Actions ◽  
Novikov Conjecture ◽  
Universal Space ◽  
The Family ◽  
Relatively Hyperbolic ◽  
Torsion Free

We study a group which is hyperbolic relative to a finite family of infinite subgroups. We show that the group satisfies the coarse Baum–Connes conjecture if each subgroup belonging to the family satisfies the coarse Baum–Connes conjecture and admits a finite universal space for proper actions. If the group is torsion-free, then it satisfies the analytic Novikov conjecture.

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