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.

Greenberg's Theorem for Quasiconvex Subgroups of Word Hyperbolic Groups

1996 ◽  
Vol 48 (6) ◽  
pp. 1224-1244 ◽  
Author(s):  
Ilya Kapovich ◽  
Hamish Short
Keyword(s):  
Finite Index ◽  
Free Groups ◽  
Hyperbolic Group ◽  
Hyperbolic Groups ◽  
Index Subgroup ◽  
Finitely Generated ◽  
Word Hyperbolic Group ◽  
Quasiconvex Subgroups ◽  
Word Hyperbolic Groups ◽  
Finite Index Subgroup

AbstractAnalogues of a theorem of Greenberg about finitely generated subgroups of free groups are proved for quasiconvex subgroups of word hyperbolic groups. It is shown that a quasiconvex subgroup of a word hyperbolic group is a finite index subgroup of only finitely many other subgroups.

scholarly journals On growth of double cosets in hyperbolic groups

2020 ◽  
Vol 30 (06) ◽  
pp. 1161-1166
Author(s):  
Rita Gitik ◽  
Eliyahu Rips
Keyword(s):  
Hyperbolic Group ◽  
Growth Function ◽  
Hyperbolic Groups ◽  
Index Formula ◽  
Infinite Index ◽  
Double Cosets ◽  
Finitely Presented Group ◽  
Quasiconvex Subgroups ◽  
Finitely Presented

Let [Formula: see text] be a hyperbolic group, [Formula: see text] and [Formula: see text] be subgroups of [Formula: see text], and [Formula: see text] be the growth function of the double cosets [Formula: see text]. We prove that the behavior of [Formula: see text] splits into two different cases. If [Formula: see text] and [Formula: see text] are not quasiconvex, we obtain that every growth function of a finitely presented group can appear as [Formula: see text]. We can even take [Formula: see text]. In contrast, for quasiconvex subgroups [Formula: see text] and [Formula: see text] of infinite index, [Formula: see text] is exponential. Moreover, there exists a constant [Formula: see text], such that [Formula: see text] for all big enough [Formula: see text], where [Formula: see text] is the growth function of the group [Formula: see text]. So, we have a clear dichotomy between the quasiconvex and non-quasiconvex case.

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 Notes on Relatively Hyperbolic Groups and Relatively Quasiconvex Subgroups

2015 ◽  
Vol 38 (1) ◽  
pp. 99-123
Author(s):  
Yoshifumi MATSUDA ◽  
Shin-ichi OGUNI ◽  
Saeko YAMAGATA
Keyword(s):  
Hyperbolic Groups ◽  
Relatively Hyperbolic Groups ◽  
Quasiconvex Subgroups ◽  
Relatively Hyperbolic

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.

scholarly journals Proper isometric actions of hyperbolic groups on -spaces

2013 ◽  
Vol 149 (5) ◽  
pp. 773-792 ◽  
Author(s):  
Bogdan Nica
Keyword(s):  
Invariant Measure ◽  
Cohomology Group ◽  
Hyperbolic Group ◽  
Hyperbolic Groups ◽  
Isometric Action ◽  
Isometric Actions

AbstractWe show that every non-elementary hyperbolic group $\G $ admits a proper affine isometric action on $L^p(\bd \G \times \bd \G )$, where $\bd \G $ denotes the boundary of $\G $ and $p$ is large enough. Our construction involves a $\G $-invariant measure on $\bd \G \times \bd \G $ analogous to the Bowen–Margulis measure from the ${\rm CAT}(-1)$ setting, as well as a geometric, Busemann-type cocycle. We also deduce that $\G $ admits a proper affine isometric action on the first $\ell ^p$-cohomology group $H^1_{(p)}(\G )$ for large enough $p$.

Sign in / Sign up

Export Citation Format

Share Document