scholarly journals ON A GENERALIZATION OF DEHN'S ALGORITHM

2008 ◽  
Vol 18 (07) ◽  
pp. 1137-1177 ◽  
Author(s):  
OLIVER GOODMAN ◽  
MICHAEL SHAPIRO
Keyword(s):  
Finite Groups ◽  
Nilpotent Groups ◽  
Hyperbolic Groups ◽  
Fundamental Groups ◽  
Closure Properties ◽  
Relatively Hyperbolic Groups ◽  
Geometrically Finite ◽  
Geometrically Finite Groups ◽  
Relatively Hyperbolic ◽  
Infinite Subgroup

Viewing Dehn's algorithm as a rewriting system, we generalize to allow an alphabet containing letters which do not necessarily represent group elements. This extends the class of groups for which the algorithm solves the word problem to include finitely generated nilpotent groups, many relatively hyperbolic groups including geometrically finite groups and fundamental groups of certain geometrically decomposable 3-manifolds. The class has several nice closure properties. We also show that if a group has an infinite subgroup and one of exponential growth, and they commute, then it does not admit such an algorithm. We dub these Cannon's algorithms.

scholarly journals Novikov Conjectures and Relative Hyperbolicity

1999 ◽  
Vol 85 (2) ◽  
pp. 169 ◽  
Author(s):  
Boris Goldfarb
Keyword(s):  
Symmetric Spaces ◽  
Large Family ◽  
Hyperbolic Groups ◽  
Fundamental Groups ◽  
Arithmetic Groups ◽  
Relatively Hyperbolic Groups ◽  
Rank One ◽  
Higher Rank ◽  
Relatively Hyperbolic ◽  
Relative Hyperbolicity

We consider a class of relatively hyperbolic groups in the sense of Gromov and use an argument modeled after Carlsson-Pedersen to prove Novikov conjectures for these groups. This proof is related to [16,17] which dealt with arithmetic lattices in rank one symmetric spaces and some other arithmetic groups of higher rank. Here whe view the rank one lattices in this different larger context of relativve hyperbolicity which also inclues fundamental groups of pinched hyperbolic manifolds. Another large family of groups from this class is produced using combinatorial hyperbolization techniques.

scholarly journals Purely exponential growth of cusp-uniform actions

2017 ◽  
Vol 39 (3) ◽  
pp. 795-831
Author(s):  
WEN-YUAN YANG
Keyword(s):  
Exponential Growth ◽  
Main Tool ◽  
Growth Function ◽  
Countable Group ◽  
Hyperbolic Groups ◽  
Growth Type ◽  
Relatively Hyperbolic Groups ◽  
Geometrically Finite ◽  
Relatively Hyperbolic ◽  
Gromov Boundary

Suppose that a countable group$G$admits a cusp-uniform action on a hyperbolic space$(X,d)$such that$G$is of divergent type. The main result of the paper is characterizing the purely exponential growth type of the orbit growth function by a condition introduced by Dal’bo, Otal and Peigné [Séries de Poincaré des groupes géométriquement finis.Israel J. Math.118(3) (2000), 109–124]. For geometrically finite Cartan–Hadamard manifolds with pinched negative curvature, this condition ensures the finiteness of Bowen–Margulis–Sullivan measures. In this case, our result recovers a theorem of Roblin (in a coarse form). Our main tool is the Patterson–Sullivan measures on the Gromov boundary of$X$, and a variant of the Sullivan shadow lemma called the partial shadow lemma. This allows us to prove that the purely exponential growth of either cones, or partial cones or horoballs is also equivalent to the Dal’bo–Otal–Peigné condition. These results are used further in a paper by the present author [W. Yang, Patterson–Sullivan measures and growth of relatively hyperbolic groups.Preprint, 2013,arXiv:1308.6326].

scholarly journals Asymptotically isometric metrics on relatively hyperbolic groups and marked length spectrum

2015 ◽  
Vol 07 (02) ◽  
pp. 345-359 ◽  
Author(s):  
Koji Fujiwara
Keyword(s):  
Hyperbolic Group ◽  
Length Spectrum ◽  
Hyperbolic Groups ◽  
Fundamental Groups ◽  
Relatively Hyperbolic Groups ◽  
Universal Covers ◽  
Relatively Hyperbolic Group ◽  
Marked Length Spectrum ◽  
Isospectral Problem ◽  
Relatively Hyperbolic

We prove asymptotically isometric, coarsely geodesic metrics on a toral relatively hyperbolic group are coarsely equal. The theorem applies to all lattices in SO (n, 1). This partly verifies a conjecture by Margulis. In the case of hyperbolic groups/spaces, our result generalizes a theorem by Furman and a theorem by Krat. We discuss an application to the isospectral problem for the length spectrum of Riemannian manifolds. The positive answer to this problem has been known for several cases. Most of them have hyperbolic fundamental groups. We do not solve the isospectral problem in the original sense, but prove the universal covers are (1, C)-quasi-isometric if the fundamental group is a toral relatively hyperbolic group.

Equations in Algebras

2018 ◽  
Vol 28 (08) ◽  
pp. 1517-1533 ◽  
Author(s):  
Olga Kharlampovich ◽  
Alexei Myasnikov
Keyword(s):  
Hyperbolic Groups ◽  
Fundamental Groups ◽  
Associative Algebras ◽  
Relatively Hyperbolic Groups ◽  
Transitive Groups ◽  
Right Angled Artin Groups ◽  
Graph Groups ◽  
Commutative Transitive ◽  
Relatively Hyperbolic ◽  
Torsion Free

We show that the Diophantine problem (decidability of equations) is undecidable in free associative algebras over any field and in the group algebras over any field of a wide variety of torsion free groups, including toral relatively hyperbolic groups, right-angled Artin groups, commutative transitive groups, the fundamental groups of various graph groups, etc.

scholarly journals RELATIVELY HYPERBOLIC GROUPS

2012 ◽  
Vol 22 (03) ◽  
pp. 1250016 ◽  
Author(s):  
B. H. BOWDITCH
Keyword(s):  
Hyperbolic Space ◽  
Limit Point ◽  
Hyperbolic Group ◽  
Hyperbolic Groups ◽  
Relatively Hyperbolic Groups ◽  
Geometrically Finite ◽  
Hyperbolic Graph ◽  
Finite Action ◽  
Relatively Hyperbolic Group ◽  
Relatively Hyperbolic

In this paper we develop some of the foundations of the theory of relatively hyperbolic groups as originally formulated by Gromov. We prove the equivalence of two definitions of this notion. One is essentially that of a group admitting a properly discontinuous geometrically finite action on a proper hyperbolic space, that is, such that every limit point is either a conical limit point or a bounded parabolic point. The other is that of a group which admits a cofinite action on a connected fine hyperbolic graph. We define a graph to be "fine" if there are only finitely many circuits a given length containing any given edge, and we develop some of the properties of this notion. We show how a relatively hyperbolic group can be assumed to act on a proper hyperbolic space of a particular geometric form. We define the boundary of a relatively hyperbolic group, and show that the limit set of any geometrically finite action of the group is equivariantly homeomorphic to this boundary. This generalizes a result of Tukia for geometrically finite kleinian groups. We also describe when the boundary is connected.

scholarly journals The local-to-global property for Morse quasi-geodesics

2021 ◽  
Author(s):  
Jacob Russell ◽  
Davide Spriano ◽  
Hung Cong Tran
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Type Theorem ◽  
Hyperbolic Groups ◽  
Global Property ◽  
Mapping Class ◽  
Relatively Hyperbolic Groups ◽  
Convex Cocompact ◽  
Relatively Hyperbolic ◽  
Translation Length

AbstractWe show the mapping class group, $${{\,\mathrm{CAT}\,}}(0)$$ CAT ( 0 ) groups, the fundamental groups of closed 3-manifolds, and certain relatively hyperbolic groups have a local-to-global property for Morse quasi-geodesics. This allows us to generalize combination theorems of Gitik for quasiconvex subgroups of hyperbolic groups to the stable subgroups of these groups. In the case of the mapping class group, this gives combination theorems for convex cocompact subgroups. We show a number of additional consequences of this local-to-global property, including a Cartan–Hadamard type theorem for detecting hyperbolicity locally and discreteness of translation length of conjugacy classes of Morse elements with a fixed gauge. To prove the relatively hyperbolic case, we develop a theory of deep points for local quasi-geodesics in relatively hyperbolic spaces, extending work of Hruska.

scholarly journals Splittings and automorphisms of relatively hyperbolic groups

2015 ◽  
Vol 9 (2) ◽  
pp. 599-663 ◽  
Author(s):  
Vincent Guirardel ◽  
Gilbert Levitt
Keyword(s):  
Hyperbolic Groups ◽  
Relatively Hyperbolic Groups ◽  
Relatively Hyperbolic

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 QUASI-ISOMETRIES, BOUNDARIES AND JSJ-DECOMPOSITIONS OF RELATIVELY HYPERBOLIC GROUPS

2013 ◽  
Vol 05 (04) ◽  
pp. 451-475 ◽  
Author(s):  
BRADLEY W. GROFF
Keyword(s):  
Hyperbolic Groups ◽  
Isometry Groups ◽  
Geometric Structures ◽  
Relatively Hyperbolic Groups ◽  
Jsj Decomposition ◽  
Outer Automorphisms ◽  
Jsj Decompositions ◽  
Relatively Hyperbolic

We demonstrate the quasi-isometry invariance of two important geometric structures for relatively hyperbolic groups: the coned space and the cusped space. As applications, we produce a JSJ-decomposition for relatively hyperbolic groups which is invariant under quasi-isometries and outer automorphisms, as well as a related splitting of the quasi-isometry groups of relatively hyperbolic groups.

Sign in / Sign up

Export Citation Format

Share Document