scholarly journals SOME SMALL CANCELLATION PROPERTIES OF RANDOM GROUPS

2007 ◽  
Vol 17 (01) ◽  
pp. 37-51 ◽  
Author(s):  
YANN OLLIVIER
Keyword(s):  
Hyperbolic Spaces ◽  
Density Parameter ◽  
Sharp Constant ◽  
Small Cancellation ◽  
Random Group ◽  
Small Loss ◽  
Small Cancellation Condition ◽  
Random Groups ◽  
Global Principle ◽  
Standard Presentation

We work in the density model of random groups. We prove that they satisfy an isoperimetric inequality with sharp constant 1-2d depending upon the density parameter d. This implies in particular a property generalizing the ordinary C′ small cancellation condition, which could be termed "macroscopic cancellation". This also sharpens the evaluation of the hyperbolicity constant δ. As a consequence we get that the standard presentation of a random group at density d < 1/5 satisfies the Dehn algorithm and Greendlinger's lemma, and that it does not for d > 1/5. For this we establish a version of the local-global principle for hyperbolic spaces (Cartan–Hadamard–Gromov theorem) involving arbitrarily small loss in the isoperimetric constant.

scholarly journals The free group on n generators modulo n + u random relations as n goes to infinity

2020 ◽  
Vol 2020 (762) ◽  
pp. 123-166
Author(s):  
Yuan Liu ◽  
Melanie Matchett Wood
Keyword(s):  
Free Group ◽  
Abelian Groups ◽  
Random Group ◽  
Random Groups

AbstractWe show that, as n goes to infinity, the free group on n generators, modulo {n+u} random relations, converges to a random group that we give explicitly. This random group is a non-abelian version of the random abelian groups that feature in the Cohen–Lenstra heuristics. For each n, these random groups belong to the few relator model in the Gromov model of random groups.

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.

On small cancellation theory over H.N.N. extensions

1983 ◽  
Vol 94 (1-2) ◽  
pp. 25-47 ◽  
Author(s):  
J. Perraud
Keyword(s):  
Word Problem ◽  
Finite Subset ◽  
Decision Problems ◽  
Normal Closure ◽  
Small Cancellation ◽  
Free Products ◽  
Cancellation Condition ◽  
Small Cancellation Theory ◽  
Amalgamated Free Products ◽  
Small Cancellation Condition

SynopsisSmall cancellation theory has been extended to symmetrized subsets of free products, amalgamated free products and Higman-Neumann-Neumann (H.N.N.) extensions. We though that it was possible to obtain results on decision problems if we could define small cancellation conditions for finite subsets.Sacerdote and Schupp (1974) defined the small cancellation condition C'(l/6) for symmetrized subsets of an H.N.N. extension. We define this condition for finite subsets, with the following properties:For each finite subset X, there is a symmetrized subset X1 with the same normal closure and, if X1 satisfies C'(l/6), then X satisfies C'(l/6).For some H.N.N. extensions, we can decide whether any finite subset satisfies C'(l/6), and, in this case, we can solve the word problem for the corresponding quotient.

scholarly journals GRAPH SMALL CANCELLATION THEORY APPLIED TO ALTERNATING LINK GROUPS

2012 ◽  
Vol 21 (11) ◽  
pp. 1250113
Author(s):  
RÉMI CUNÉO ◽  
HAMISH SHORT
Keyword(s):  
Small Cancellation ◽  
Cancellation Condition ◽  
Small Cancellation Theory ◽  
Link Group ◽  
Finite Sums ◽  
Small Cancellation Condition ◽  
Alternating Links ◽  
Alternating Link

We show that the Wirtinger presentation of a prime alternating link group satisfies a generalized small cancellation condition. This new version of Weinbaum's solution to the word and conjugacy problems for these groups easily extends to finite sums of alternating links.

scholarly journals Generic properties of random subgroups of a free group for general distributions

2012 ◽  
Vol DMTCS Proceedings vol. AQ,... (Proceedings) ◽  
Author(s):  
Frédérique Bassino ◽  
Cyril Nicaud ◽  
Pascal Weil
Keyword(s):  
Free Group ◽  
Random Variable ◽  
Small Cancellation ◽  
Group Presentation ◽  
Generic Properties ◽  
Cancellation Condition ◽  
Number Of Generators ◽  
International Audience ◽  
General Distributions ◽  
Small Cancellation Condition

International audience We consider a generalization of the uniform word-based distribution for finitely generated subgroups of a free group. In our setting, the number of generators is not fixed, the length of each generator is determined by a random variable with some simple constraints and the distribution of words of a fixed length is specified by a Markov process. We show by probabilistic arguments that under rather relaxed assumptions, the good properties of the uniform word-based distribution are preserved: generically (but maybe not exponentially generically), the tuple we pick is a basis of the subgroup it generates, this subgroup is malnormal and the group presentation defined by this tuple satisfies a small cancellation condition.

On T(6)-groups

1987 ◽  
Vol 102 (3) ◽  
pp. 443-451 ◽  
Author(s):  
Mohamed S. El-Mosalamy ◽  
Stephen J. Pride
Keyword(s):  
Small Cancellation ◽  
Cancellation Condition ◽  
Group Presentations ◽  
Definition Of ◽  
Small Cancellation Condition

This paper is concerned with group presentations satisfying the small cancellation condition T(6). The definition of this condition is given in §1·2, together with some examples. Before giving the definition, however, we describe (in §1·1) some material which, to a certain extent, motivated our paper. In § 1·3 we state our main theorem, which provides new solutions to the word and conjugacy problems for T(6)-groups.

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