Unsolvable Problems About Small Cancellation and Word Hyperbolic Groups

1994 ◽  
Vol 26 (1) ◽  
pp. 97-101 ◽  
Author(s):  
G. Baumslag ◽  
C. F. Miller ◽  
H. Short
Keyword(s):  
Hyperbolic Groups ◽  
Small Cancellation ◽  
Word Hyperbolic Groups ◽  
Unsolvable Problems

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 Cogrowth for group actions with strongly contracting elements

2018 ◽  
Vol 40 (7) ◽  
pp. 1738-1754 ◽  
Author(s):  
GOULNARA N. ARZHANTSEVA ◽  
CHRISTOPHER H. CASHEN
Keyword(s):  
Mapping Class Groups ◽  
Hyperbolic Spaces ◽  
Hyperbolic Groups ◽  
Mapping Class ◽  
Small Cancellation ◽  
Class Groups ◽  
Original Result ◽  
Geodesic Metric Space ◽  
Geodesic Metric ◽  
Right Angled Artin Groups

Let $G$ be a group acting properly by isometries and with a strongly contracting element on a geodesic metric space. Let $N$ be an infinite normal subgroup of $G$ and let $\unicode[STIX]{x1D6FF}_{N}$ and $\unicode[STIX]{x1D6FF}_{G}$ be the growth rates of $N$ and $G$ with respect to the pseudo-metric induced by the action. We prove that if $G$ has purely exponential growth with respect to the pseudo-metric, then $\unicode[STIX]{x1D6FF}_{N}/\unicode[STIX]{x1D6FF}_{G}>1/2$. Our result applies to suitable actions of hyperbolic groups, right-angled Artin groups and other CAT(0) groups, mapping class groups, snowflake groups, small cancellation groups, etc. This extends Grigorchuk’s original result on free groups with respect to a word metric and a recent result of Matsuzaki, Yabuki and Jaerisch on groups acting on hyperbolic spaces to a much wider class of groups acting on spaces that are not necessarily hyperbolic.

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.

scholarly journals On the geometry of burnside quotients of torsion free hyperbolic groups

2014 ◽  
Vol 24 (03) ◽  
pp. 251-345 ◽  
Author(s):  
Rémi Coulon
Keyword(s):  
Hyperbolic Groups ◽  
Geometrical Approach ◽  
Small Cancellation ◽  
Small Cancellation Theory ◽  
Burnside Groups ◽  
Torsion Free

In this paper, we detail the geometrical approach of small cancellation theory used by Delzant and Gromov to provide a new proof of the infiniteness of free Burnside groups and periodic quotients of torsion-free hyperbolic groups.

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 A combination theorem for combinatorially non-positively curved complexes of hyperbolic groups

2020 ◽  
pp. 1-33
Author(s):  
ALEXANDRE MARTIN ◽  
DAMIAN OSAJDA
Keyword(s):  
Positive Curvature ◽  
Weak Form ◽  
Hyperbolic Groups ◽  
Combinatorial Property ◽  
Small Cancellation ◽  
Large Classes ◽  
Positively Curved ◽  
Gromov Boundary

Abstract We prove a combination theorem for hyperbolic groups, in the case of groups acting on complexes displaying combinatorial features reminiscent of non-positive curvature. Such complexes include for instance weakly systolic complexes and C'(1/6) small cancellation polygonal complexes. Our proof involves constructing a potential Gromov boundary for the resulting groups and analyzing the dynamics of the action on the boundary in order to use Bowditch’s characterisation of hyperbolicity. A key ingredient is the introduction of a combinatorial property that implies a weak form of non-positive curvature, and which holds for large classes of complexes. As an application, we study the hyperbolicity of groups obtained by small cancellation over a graph of hyperbolic groups.

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