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.

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.

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.

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.

VERIFYING THAT A GROUP IS VIRTUALLY FREE

1991 ◽  
Vol 01 (03) ◽  
pp. 339-351
Author(s):  
ROBERT H. GILMAN
Keyword(s):  
Finite Index ◽  
Free Subgroup ◽  
Universal Group ◽  
Finitely Presented Groups ◽  
Finite Presentation ◽  
Finitely Presented

This paper is concerned with computation in finitely presented groups. We discuss a procedure for showing that a finite presentation presents a group with a free subgroup of finite index, and we give methods for solving various problems in such groups. Our procedure works by constructing a particular kind of partial groupoid whose universal group is isomorphic to the group presented. When the procedure succeeds, the partial groupoid can be used as an aid to computation in the group.

Decision problems concerning S-arithmetic groups

1985 ◽  
Vol 50 (3) ◽  
pp. 743-772 ◽  
Author(s):  
Fritz Grunewald ◽  
Daniel Segal
Keyword(s):  
Additive Group ◽  
Nilpotent Groups ◽  
Isomorphism Problem ◽  
Algorithmic Problem ◽  
Finitely Generated ◽  
Finitely Presented Groups ◽  
Finite Dimensional ◽  
Associative Rings ◽  
Finitely Presented ◽  
Torsion Free

This paper is a continuation of our previous work in [12]. The results, and some applications, have been described in the announcement [13]; it may be useful to discuss here, a little more fully, the nature and purpose of this work.We are concerned basically with three kinds of algorithmic problem: (1) isomorphism problems, (2) “orbit problems”, and (3) “effective generation”.(1) Isomorphism problems. Here we have a class of algebraic objects of some kind, and ask: is there a uniform algorithm for deciding whether two arbitrary members of are isomorphic? In most cases, the answer is no: no such algorithm exists. Indeed this has been one of the most notable applications of methods of mathematical logic in algebra (see [26, Chapter IV, §4] for the case where is the class of all finitely presented groups). It turns out, however, that when consists of objects which are in a certain sense “finite-dimensional”, then the isomorphism problem is indeed algorithmically soluble. We gave such algorithms in [12] for the following cases: = {finitely generated nilpotent groups}; = {(not necessarily associative) rings whose additive group is finitely generated}; = {finitely Z-generated modules over a fixed finitely generated ring}.Combining the methods of [12] with his own earlier work, Sarkisian has obtained analogous results with the integers replaced by the rationals: in [20] and [21] he solves the isomorphism problem for radicable torsion-free nilpotent groups of finite rank and for finite-dimensional Q-algebras.

scholarly journals A computer-aided analysis of some finitely presented groups

1992 ◽  
Vol 53 (3) ◽  
pp. 369-376 ◽  
Author(s):  
M. F. Newman ◽  
E. A. O'Brien
Keyword(s):  
Finite Group ◽  
Computer Programs ◽  
Free Products ◽  
Finitely Presented Groups ◽  
Cyclic Groups ◽  
Group Presentations ◽  
Computer Aided Analysis ◽  
Computer Aided ◽  
Finitely Presented

AbstractWe answer some questions which arise from a recent paper of Campbell, Heggie, Robertson and Thomas on one-relator free products of two cyclic groups. In the process we show how publicly accessible computer programs can be used to help answer questions about finite group presentations.

scholarly journals Direct products and properly 3-realisable groups

2004 ◽  
Vol 70 (2) ◽  
pp. 199-205 ◽  
Author(s):  
Manuel Cárdenas ◽  
Francisco F. Lasheras ◽  
Ranja Roy
Keyword(s):  
Homotopy Type ◽  
Universal Cover ◽  
Hyperbolic Groups ◽  
Direct Products ◽  
Finitely Presented Groups ◽  
Finitely Presented Group ◽  
Manifold With Boundary ◽  
Proper Homotopy ◽  
Finitely Presented

In this paper, we show that the direct of infinite finitely presented groups is always properly 3-realisable. We also show that classical hyperbolic groups are properly 3-realisable. We recall that a finitely presented group G is said to be properly 3-realisable if there exists a compact 2-polyhedron K with π1 (K) ≅ G and whose universal cover K̃ has the proper homotopy type of a (p.1.) 3-manifold with boundary. The question whether or not every finitely presented is properly 3-realisable remains open.

scholarly journals COMPUTATION IN WORD-HYPERBOLIC GROUPS

2001 ◽  
Vol 11 (04) ◽  
pp. 467-487 ◽  
Author(s):  
DAVID B. A. EPSTEIN ◽  
DEREK F. HOLT
Keyword(s):  
Cayley Graph ◽  
Difficult Problem ◽  
Hyperbolic Groups ◽  
Maximum Width ◽  
Finitely Presented Group ◽  
Practical Algorithms ◽  
Finite State ◽  
Geodesic Triangles ◽  
Word Hyperbolic Groups ◽  
Finitely Presented

We describe two practical algorithms for computing with word-hyperbolic groups, both of which we have implemented. The first is a method for estimating the maximum width, if it exists, of geodesic bigons in the Cayley graph of a finitely presented group G. Our procedure will terminate if and only this maximum width exists, and it has been proved by Papasoglu that this is the case if and only if G is word-hyperbolic. So the algorithm amounts to a method of verifying the property of word-hyperbolicity of G. The aim of the second algorithm is to compute the thinness constant for geodesic triangles in the Cayley graph of G. This seems to be a much more difficult problem, but our implementation does succeed with straightforward examples. Both algorithms involve substantial computations with finite state automata.

scholarly journals A periodicity theorem for acylindrically hyperbolic groups

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Oleg Bogopolski
Keyword(s):  
Free Groups ◽  
Hyperbolic Groups ◽  
Finitely Generated ◽  
Finitely Presented Groups ◽  
Periodicity Theorem ◽  
Finitely Presented

AbstractWe generalize a well-known periodicity lemma from the case of free groups to the case of acylindrically hyperbolic groups. This generalization has been used to describe solutions of certain equations in acylindrically hyperbolic groups and to characterize verbally closed finitely generated acylindrically hyperbolic subgroups of finitely presented groups.

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