small cancellation
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2021 ◽  
Vol 13 (1) ◽  
pp. 79-102
Author(s):  
Stéphane Lamy ◽  
Anne Lonjou


10.5802/mrr.6 ◽  
2021 ◽  
Vol 2 ◽  
pp. 1-14
Author(s):  
Agatha Atkarskaya ◽  
Alexei Kanel-Belov ◽  
Eugene Plotkin ◽  
Eliyahu Rips
Keyword(s):  


Author(s):  
Arye Juhász

It is conjectured that an irreducible Artin group which is of infinite type has trivial center. The conjecture is known to be true for two-dimensional Artin groups and for a few other types of Artin groups. In this work, we show that the conjecture holds true for Artin groups which satisfy a condition stronger than being of infinite type. We use small cancellation theory of relative presentations.



Author(s):  
Martín Axel Blufstein ◽  
Elías Gabriel Minian ◽  
Iván Sadofschi Costa

We present a metric condition $\TTMetric$ which describes the geometry of classical small cancellation groups and applies also to other known classes of groups such as two-dimensional Artin groups. We prove that presentations satisfying condition $\TTMetric$ are diagrammatically reducible in the sense of Sieradski and Gersten. In particular, we deduce that the standard presentation of an Artin group is aspherical if and only if it is diagrammatically reducible. We show that, under some extra hypotheses, $\TTMetric$ -groups have quadratic Dehn functions and solvable conjugacy problem. In the spirit of Greendlinger's lemma, we prove that if a presentation P = 〈X| R〉 of group G satisfies conditions $\TTMetric -C'(\frac {1}{2})$ , the length of any nontrivial word in the free group generated by X representing the trivial element in G is at least that of the shortest relator. We also introduce a strict metric condition $\TTMetricStrict$ , which implies hyperbolicity.



Author(s):  
DOMINIK GRUBER ◽  
ALESSANDRO SISTO
Keyword(s):  

Abstract We show that Gromov’s monsters arising from i.i.d. random labellings of expanders (that we call random Gromov’s monsters) have linear divergence along a subsequence, so that in particular they do not contain Morse quasigeodesics, and they are not quasi-isometric to Gromov’s monsters arising from graphical small cancellation labellings of expanders. Moreover, by further studying the divergence function, we show that there are uncountably many quasi-isometry classes of random Gromov’s monsters.



2020 ◽  
Vol volume 12, issue 2 ◽  
Author(s):  
Alex Bishop ◽  
Michal Ferov

Small cancellation groups form an interesting class with many desirable properties. It is a well-known fact that small cancellation groups are generic; however, all previously known results of their genericity are asymptotic and provide no information about "small" group presentations. In this note, we give closed-form formulas for both lower and upper bounds on the density of small cancellation presentations, and compare our results with experimental data. Comment: 18 pages, 12 figures



Author(s):  
Carolyn R. Abbott ◽  
David Hume


Author(s):  
ALEXANDRE MARTIN ◽  
DAMIAN OSAJDA

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.



2020 ◽  
Vol 225 (1) ◽  
pp. 159-191
Author(s):  
Damian Osajda


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