scholarly journals Quasiconvexity and Amalgams

1997 ◽  
Vol 07 (06) ◽  
pp. 771-811 ◽  
Author(s):  
Ilya Kapovich
Keyword(s):  
Cyclic Subgroup ◽  
Hyperbolic Group ◽  
Hyperbolic Surface ◽  
Kleinian Groups ◽  
Hyperbolic Groups ◽  
3 Dimensional ◽  
Word Hyperbolic Group ◽  
Amalgamated Free Product ◽  
One Relator Groups ◽  
Word Hyperbolic Groups

We obtain a criterion for quasiconvexity of a subgroup of an amalgamated free product of two word hyperbolic groups along a virtually cyclic subgroup. The result provides a method of constructing new word hyperbolic group in class (Q), that is such that all their finitely generated subgroups are quasiconvex. It is known that free groups, hyperbolic surface groups and most 3-dimensional Kleinian groups have property (Q). We also give some applications of our results to one-relator groups and exponential groups.

Amalgamated Products and the Howson Property

1997 ◽  
Vol 40 (3) ◽  
pp. 330-340 ◽  
Author(s):  
Ilya Kapovich
Keyword(s):  
Hyperbolic Group ◽  
Hyperbolic Groups ◽  
Fundamental Groups ◽  
Finitely Generated ◽  
Word Hyperbolic Group ◽  
Amalgamated Products ◽  
Word Hyperbolic Groups ◽  
Free Word ◽  
Torsion Free ◽  
Maximal Cyclic Subgroup

AbstractWe show that if A is a torsion-free word hyperbolic group which belongs to class (Q), that is all finitely generated subgroups of A are quasiconvex in A, then any maximal cyclic subgroup U of A is a Burns subgroup of A. This, in particular, implies that if B is a Howson group (that is the intersection of any two finitely generated subgroups is finitely generated) then A *UB, ⧼A, t | Ut = V⧽ are also Howson groups. Finitely generated free groups, fundamental groups of closed hyperbolic surfaces and some interesting 3-manifold groups are known to belong to class (Q) and our theorem applies to them. We also describe a large class of word hyperbolic groups which are not Howson.

scholarly journals Quasiregular self-mappings of manifolds and word hyperbolic groups

2007 ◽  
Vol 143 (6) ◽  
pp. 1613-1622 ◽  
Author(s):  
Martin Bridson ◽  
Aimo Hinkkanen ◽  
Gaven Martin
Keyword(s):  
Hyperbolic Group ◽  
Quasiregular Mapping ◽  
Hyperbolic Groups ◽  
Quasiregular Mappings ◽  
Word Hyperbolic Group ◽  
Word Hyperbolic Groups ◽  
Mostow Rigidity ◽  
Negative Sectional Curvature ◽  
Free Word ◽  
Torsion Free

AbstractAn extension of a result of Sela shows that if Γ is a torsion-free word hyperbolic group, then the only homomorphisms Γ→Γ with finite-index image are the automorphisms. It follows from this result and properties of quasiregular mappings, that if M is a closed Riemannian n-manifold with negative sectional curvature ($n\neq 4$), then every quasiregular mapping f:M→M is a homeomorphism. In the constant-curvature case the dimension restriction is not necessary and Mostow rigidity implies that f is homotopic to an isometry. This is to be contrasted with the fact that every such manifold admits a non-homeomorphic light open self-mapping. We present similar results for more general quotients of hyperbolic space and quasiregular mappings between them. For instance, we establish that besides covering projections there are no π1-injective proper quasiregular mappings f:M→N between hyperbolic 3-manifolds M and N with non-elementary fundamental group.

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.

THE LINEARITY OF THE CONJUGACY PROBLEM IN WORD-HYPERBOLIC GROUPS

2006 ◽  
Vol 16 (02) ◽  
pp. 287-305 ◽  
Author(s):  
DAVID EPSTEIN ◽  
DEREK HOLT
Keyword(s):  
Normal Form ◽  
Theoretical Result ◽  
Linear Time ◽  
Hyperbolic Group ◽  
Conjugacy Problem ◽  
Hyperbolic Groups ◽  
Constant Time ◽  
Word Hyperbolic Group ◽  
Group A ◽  
Word Hyperbolic Groups

The main result proved in this paper is that the conjugacy problem in word-hyperbolic groups is solvable in linear time. This is using a standard RAM model of computation, in which basic arithmetical operations on integers are assumed to take place in constant time. The constants involved in the linear time solution are all computable explicitly. We also give a proof of the result of Mike Shapiro that in a word-hyperbolic group a word in the generators can be transformed into short-lex normal form in linear time. This is used in the proof of our main theorem, but is a significant theoretical result of independent interest, which deserves to be in the literature. Previously the best known result was a quadratic estimate.

Hyperbolicity of some 2-generated one-relator groups

2002 ◽  
Vol 12 (4) ◽  
Author(s):  
N. B. Bezverkhnyaya
Keyword(s):  
Basic Research ◽  
Hyperbolic Groups ◽  
One Relator Groups ◽  
Basic Research Grant ◽  
Word Hyperbolic Groups ◽  
Research Grant

AbstractWe give the description of word hyperbolic groups of the form(a, b, t; tThe research was supported by the Russian Foundation for Basic Research, grant 00-01-00767.

scholarly journals Surface groups within Baumslag doubles

2011 ◽  
Vol 54 (1) ◽  
pp. 91-97 ◽  
Author(s):  
Benjamin Fine ◽  
Gerhard Rosenberger
Keyword(s):  
Fundamental Group ◽  
Hyperbolic Group ◽  
Sufficient Conditions ◽  
Surface Group ◽  
Orientable Surface ◽  
Hyperbolic Surface ◽  
Surface Groups ◽  
Word Hyperbolic Group ◽  
Genus 2

AbstractA conjecture of Gromov states that a one-ended word-hyperbolic group must contain a subgroup that is isomorphic to the fundamental group of a closed hyperbolic surface. Recent papers by Gordon and Wilton and by Kim and Wilton give sufficient conditions for hyperbolic surface groups to be embedded in a hyperbolic Baumslag double G. Using Nielsen cancellation methods based on techniques from previous work by the second author, we prove that a hyperbolic orientable surface group of genus 2 is embedded in a hyperbolic Baumslag double if and only if the amalgamated word W is a commutator: that is, W = [U, V] for some elements U, V ∈ F. Furthermore, a hyperbolic Baumslag double G contains a non-orientable surface group of genus 4 if and only if W = X2Y2 for some X, Y ∈ F. G can contain no non-orientable surface group of smaller genus.

scholarly journals THE CONJUGACY PROBLEM IN HYPERBOLIC GROUPS FOR FINITE LISTS OF GROUP ELEMENTS

2013 ◽  
Vol 23 (05) ◽  
pp. 1127-1150 ◽  
Author(s):  
D. J. BUCKLEY ◽  
DEREK F. HOLT
Keyword(s):  
Upper Bound ◽  
Hyperbolic Group ◽  
Conjugacy Problem ◽  
Hyperbolic Groups ◽  
Generating Set ◽  
Running Time ◽  
Word Hyperbolic Group ◽  
Practical Algorithm

Let G be a word-hyperbolic group with given finite generating set, for which various standard structures and constants have been pre-computed. An (non-practical) algorithm is described that, given as input two lists A and B, each composed of m words in the generators and their inverses, determines whether or not the lists are conjugate in G, and returns a conjugating element, should one exist. The algorithm runs in time O(mμ), where μ is an upper bound on the length of elements in the two lists. Similarly, an algorithm is outlined that computes generators of the centralizer of A, with the same bound on running time.

scholarly journals Combable functions, quasimorphisms, and the central limit theorem

2009 ◽  
Vol 30 (5) ◽  
pp. 1343-1369 ◽  
Author(s):  
DANNY CALEGARI ◽  
KOJI FUJIWARA
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Word Length ◽  
Hyperbolic Groups ◽  
Finite State Automaton ◽  
List Type ◽  
Generating Sets ◽  
Finite State ◽  
Word Hyperbolic Groups

AbstractA function on a discrete group is weakly combable if its discrete derivative with respect to a combing can be calculated by a finite-state automaton. A weakly combable function is bicombable if it is Lipschitz in both the left- and right-invariant word metrics. Examples of bicombable functions on word-hyperbolic groups include:(1)homomorphisms to ℤ;(2)word length with respect to a finite generating set;(3)most known explicit constructions of quasimorphisms (e.g. the Epstein–Fujiwara counting quasimorphisms).We show that bicombable functions on word-hyperbolic groups satisfy acentral limit theorem: if$\overline {\phi }_n$is the value of ϕ on a random element of word lengthn(in a certain sense), there areEandσfor which there is convergence in the sense of distribution$n^{-1/2}(\overline {\phi }_n - nE) \to N(0,\sigma )$, whereN(0,σ) denotes the normal distribution with standard deviationσ. As a corollary, we show that ifS1andS2are any two finite generating sets forG, there is an algebraic numberλ1,2depending onS1andS2such that almost every word of lengthnin theS1metric has word lengthn⋅λ1,2in theS2metric, with error of size$O(\sqrt {n})$.

scholarly journals Metric Characterizations of Superreflexivity in Terms of Word Hyperbolic Groups and Finite Graphs

2014 ◽  
Vol 2 (1) ◽  
Author(s):  
Mikhail Ostrovskii
Keyword(s):  
Hilbert Space ◽  
Hyperbolic Groups ◽  
Finite Graphs ◽  
Word Hyperbolic Groups

Abstract We show that superreflexivity can be characterized in terms of bilipschitz embeddability of word hyperbolic groups.We compare characterizations of superrefiexivity in terms of diamond graphs and binary trees.We show that there exist sequences of series-parallel graphs of increasing topological complexitywhich admit uniformly bilipschitz embeddings into a Hilbert space, and thus do not characterize superrefiexivity.

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