scholarly journals Hyperbolic Groups That Are Not Commensurably Co-Hopfian

2020 ◽  
Author(s):  
Emily Stark ◽  
Daniel J Woodhouse
Keyword(s):  
Fundamental Group ◽  
Hyperbolic Group ◽  
Hyperbolic Groups ◽  
Torsion Free ◽  
Simple Surface

Abstract Sela proved that every torsion-free one-ended hyperbolic group is co-Hopfian. We prove that there exist torsion-free one-ended hyperbolic groups that are not commensurably co-Hopfian. In particular, we show that the fundamental group of every simple surface amalgam is not commensurably co-Hopfian.

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.

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.

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 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$h\in H$.

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 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.

CONJUGACY OF FINITE SUBSETS IN HYPERBOLIC GROUPS

2005 ◽  
Vol 15 (04) ◽  
pp. 725-756 ◽  
Author(s):  
MARTIN R. BRIDSON ◽  
JAMES HOWIE
Keyword(s):  
Linear Function ◽  
Hyperbolic Group ◽  
Conjugacy Problem ◽  
Time Algorithm ◽  
Hyperbolic Groups ◽  
Finitely Presented Groups ◽  
Curved Spaces ◽  
Negatively Curved ◽  
Finitely Presented ◽  
Torsion Free

There is a quadratic-time algorithm that determines conjugacy between finite subsets in any torsion-free hyperbolic group. Moreover, in any k-generator, δ-hyperbolic group Γ, if two finite subsets A and B are conjugate, then x-1 Ax = B for some x ∈ Γ with ǁxǁ less than a linear function of max {ǁγǁ : γ ∈ A ∪ B}. (The coefficients of this linear function depend only on k and δ.) These results have implications for group-based cryptography and the geometry of homotopies in negatively curved spaces. In an appendix, we give examples of finitely presented groups in which the conjugacy problem for elements is soluble but the conjugacy problem for finite lists is not.

scholarly journals Effective construction of covers of canonical Hom-diagrams for equations over torsion-free hyperbolic groups

2019 ◽  
Vol 11 (2) ◽  
pp. 83-101
Author(s):  
Olga Kharlampovich ◽  
Alexei Myasnikov ◽  
Alexander Taam
Keyword(s):  
Hyperbolic Group ◽  
Finite System ◽  
Hyperbolic Groups ◽  
System Of Equations ◽  
Limit Groups ◽  
Canonical Solution ◽  
Finitely Generated Group ◽  
Coordinate Group ◽  
Torsion Free

Abstract We show that, given a finitely generated group G as the coordinate group of a finite system of equations over a torsion-free hyperbolic group Γ, there is an algorithm which constructs a cover of a canonical solution diagram. The diagram encodes all homomorphisms from G to Γ as compositions of factorizations through Γ-NTQ groups and canonical automorphisms of the corresponding NTQ-subgroups. We also give another characterization of Γ-limit groups as iterated generalized doubles over Γ.

scholarly journals Effective embedding of residually hyperbolic groups into direct products of extensions of centralizers

2013 ◽  
Vol 16 (5) ◽  
Author(s):  
Olga Kharlampovich ◽  
Jeremy Macdonald
Keyword(s):  
Hyperbolic Group ◽  
Hyperbolic Groups ◽  
Direct Products ◽  
Torsion Free

Abstract.For any torsion-free hyperbolic group Γ and any group

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