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Density of small-cancellation group presentations

Michal Ferov, University of Newcastle (Australia)
Tuesday, October 29, 2019 - 1:30pm to 3:20pm
PDL C-401

Abstract: The original motivation for studying groups was that they capture the structure of symmetries of mathematical objects. It is therefore natural to ask certain questions, for example we might want to be able to determine whether two elements are different or whether two elements are conjugate. Unfortunately, when given only a presentation without any other knowledge, there is no uniform algorithm to answer those question. What’s even worse, there is no uniform algorithm to decide whether a certain problem is decidable. However, there classes of of groups in which some problems can be answered efficiently. One such class are the hyperbolic groups. Due to a classical result of Gromov, the property of being hyperbolic is strongly generic, i.e. a random presentation will with overwhelming probability produce a hyperbolic group. There is a catch though - deciding whether a group is hyperbolic is an unsolvable problem. Not all hope is lost - large subclass of hyperbolic groups can be given by small-cancellation presentation - a combinatorial property of a presentation that can be easily checked by hand. It was proved by Azhantseva and Olshanski that small cancellation presentations are strongly generic as well. We expand on their work by giving a closed form for a lower bound on the probability that a finite presentation has small cancellation and we back our estimates by experimental data.

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