The Kadison-Kaplansky conjecture for word-hyperbolic groups

2002 ◽  
Vol 149 (1) ◽  
pp. 153-194 ◽  
Author(s):  
Michael Puschnigg
Keyword(s):  
Hyperbolic Groups ◽  
Word Hyperbolic Groups ◽  
Kaplansky Conjecture

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.

scholarly journals SOLVING THE WORD PROBLEM IN REAL TIME

2001 ◽  
Vol 63 (3) ◽  
pp. 623-639 ◽  
Author(s):  
DEREK F. HOLT ◽  
SARAH REES
Keyword(s):  
Recent Result ◽  
Real Time ◽  
Word Problem ◽  
Turing Machine ◽  
Linear Time ◽  
Nilpotent Groups ◽  
Hyperbolic Groups ◽  
Turing Machines ◽  
Geometrically Finite ◽  
Word Hyperbolic Groups

The paper is devoted to the study of groups whose word problem can be solved by a Turing machine which operates in real time. A recent result of the first author for word hyperbolic groups is extended to prove that under certain conditions the generalised Dehn algorithms of Cannon, Goodman and Shapiro, which clearly run in linear time, can be programmed on real-time Turing machines. It follows that word-hyperbolic groups, finitely generated nilpotent groups and geometrically finite hyperbolic groups all have real-time word problems.

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.

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 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 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 Engulfing in word-hyperbolic groups

2002 ◽  
Vol 2 (2) ◽  
pp. 743-755 ◽  
Author(s):  
Graham A Niblo ◽  
Ben T Williams
Keyword(s):  
Hyperbolic Groups ◽  
Word Hyperbolic Groups
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