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.

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.

AN UPPER BOUND FOR THE GROWTH OF CONJUGACY CLASSES IN TORSION-FREE WORD HYPERBOLIC GROUPS

2004 ◽  
Vol 14 (04) ◽  
pp. 395-401 ◽  
Author(s):  
MICHEL COORNAERT ◽  
GERHARD KNIEPER
Keyword(s):  
Upper Bound ◽  
Conjugacy Classes ◽  
Hyperbolic Groups ◽  
Word Hyperbolic Groups ◽  
Free Word ◽  
Torsion Free

We give a new upper bound for the growth of primitive conjugacy classes in torsion-free word hyperbolic groups.

scholarly journals Torsion-free word hyperbolic groups are noncommutatively slender

2016 ◽  
Vol 26 (07) ◽  
pp. 1467-1482 ◽  
Author(s):  
Samuel M. Corson
Keyword(s):  
Free Subgroup ◽  
Hyperbolic Groups ◽  
Topological Groups ◽  
Free Products ◽  
Finitely Presented Groups ◽  
Word Hyperbolic Groups ◽  
Hawaiian Earring ◽  
Free Word ◽  
Finitely Presented ◽  
Torsion Free

In this paper, we prove the claim given in the title. A group [Formula: see text] is noncommutatively slender if each map from the fundamental group of the Hawaiian Earring to [Formula: see text] factors through projection to a canonical free subgroup. Higman, in his seminal 1952 paper [Unrestricted free products and varieties of topological groups, J. London Math. Soc. 27 (1952) 73–81], proved that free groups are noncommutatively slender. Such groups were first defined by Eda in [Free [Formula: see text]-products and noncommutatively slender groups, J. Algebra 148 (1992) 243–263]. Eda has asked which finitely presented groups are noncommutatively slender. This result demonstrates that random finitely presented groups in the few-relator sense are noncommutatively slender.

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.

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.

On the rank of quotients of hyperbolic groups

2013 ◽  
Vol 16 (5) ◽  
Author(s):  
Richard Weidmann
Keyword(s):  
Hyperbolic Group ◽  
Hyperbolic Groups ◽  
Normal Closure ◽  
Torsion Free

Abstract.We show that the rank does not decrease if one passes from a torsion-free locally quasi-convex hyperbolic group to the quotient by the normal closure of certain high powered elements. An argument provided by Ilya Kapovich further shows that the quasiconvexity assumption cannot be dropped without adding other assumptions.

scholarly journals Hyperbolic groups with boundary an n-dimensional Sierpinski space

2019 ◽  
Vol 11 (01) ◽  
pp. 233-247
Author(s):  
Jean-François Lafont ◽  
Bena Tshishiku
Keyword(s):  
Fundamental Group ◽  
Negative Curvature ◽  
Hyperbolic Group ◽  
Hyperbolic Groups ◽  
Manifolds With Boundary ◽  
Geodesic Boundary ◽  
Totally Geodesic ◽  
Visual Boundary ◽  
Aspherical Manifolds ◽  
Torsion Free

For [Formula: see text], we show that if [Formula: see text] is a torsion-free hyperbolic group whose visual boundary [Formula: see text] is an [Formula: see text]-dimensional Sierpinski space, then [Formula: see text] for some aspherical [Formula: see text]-manifold [Formula: see text] with non-empty boundary. Concerning the converse, we construct, for each [Formula: see text], examples of aspherical manifolds with boundary, whose fundamental group [Formula: see text] is hyperbolic, but with visual boundary [Formula: see text] not homeomorphic to [Formula: see text]. Our examples even support (metric) negative curvature, and have totally geodesic boundary.

scholarly journals ON ENDOMORPHISMS OF TORSION-FREE HYPERBOLIC GROUPS

2011 ◽  
Vol 21 (08) ◽  
pp. 1415-1446 ◽  
Author(s):  
O. BOGOPOLSKI ◽  
E. VENTURA
Keyword(s):  
Hyperbolic Group ◽  
Hyperbolic Groups ◽  
Generating Set ◽  
Inner Automorphism ◽  
Torsion Free

Let H be a torsion-free δ-hyperbolic group with respect to a finite generating set S. From the main result in the paper, Theorem 1.2, we deduce the following two corollaries. First, we show that there exists a computable constant [Formula: see text] such that, for any endomorphism φ of H, if φ(h) is conjugate to h for every element h ∈ H of length up to [Formula: see text], then φ is an inner automorphism. Second, we show a mixed (conjugate/non-conjugate) version of the classical Whitehead problem for tuples is solvable in torsion-free hyperbolic groups.

scholarly journals Local rigidity for hyperbolic groups with Sierpiński carpet boundaries

2014 ◽  
Vol 150 (11) ◽  
pp. 1928-1938 ◽  
Author(s):  
Sergei Merenkov
Keyword(s):  
Hyperbolic Group ◽  
Hyperbolic Groups ◽  
Sierpinski Carpet ◽  
Limit Sets ◽  
Local Rigidity ◽  
Boundary Map ◽  
Complementary Component ◽  
Finite Index Subgroup ◽  
Sierpiński Carpet ◽  
Torsion Free

AbstractLet$G$and$\tilde{G}$be Kleinian groups whose limit sets$S$and$\tilde{S}$, respectively, are homeomorphic to the standard Sierpiński carpet, and such that every complementary component of each of$S$and$\tilde{S}$is a round disc. We assume that the groups$G$and$\tilde{G}$act cocompactly on triples on their respective limit sets. The main theorem of the paper states that any quasiregular map (in a suitably defined sense) from an open connected subset of$S$to$\tilde{S}$is the restriction of a Möbius transformation that takes$S$onto$\tilde{S}$, in particular it has no branching. This theorem applies to the fundamental groups of compact hyperbolic 3-manifolds with non-empty totally geodesic boundaries. One consequence of the main theorem is the following result. Assume that$G$is a torsion-free hyperbolic group whose boundary at infinity$\partial _{\infty }G$is a Sierpiński carpet that embeds quasisymmetrically into the standard 2-sphere. Then there exists a group$H$that contains$G$as a finite index subgroup and such that any quasisymmetric map$f$between open connected subsets of$\partial _{\infty }G$is the restriction of the induced boundary map of an element$h\in H$.

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