On random presentations with fixed relator length
Abstract
The standard (n, k, d) model of random groups is a model where the relators are chosen randomly from the set of cyclically reduced words of length k on an n-element generating set. Gromov’s density model of random groups considers the case where n is fixed, and k tends to infinity. We instead fix k, and let n tend to infinity. We prove that for all k ≥ 2 at density d > 1/2 a random group in this model is trivial or cyclic of order two, whilst for d < 1 such 2 a random group is infinite and hyperbolic. In addition we show that for d < 1/k such a random k group is free, and that this threshold is sharp. These extend known results for the triangular (k = 3) and square (k = 4) models of random groups.
Citation
Ashcroft , C & Roney-Dougal , C M 2020 , ' On random presentations with fixed relator length ' , Communications in Algebra , vol. Latest Articles . https://doi.org/10.1080/00927872.2019.1710161
Publication
Communications in Algebra
Status
Peer reviewed
ISSN
0092-7872Type
Journal article
Rights
Copyright © 2020 Taylor & Francis Group, LLC. This work has been made available online in accordance with publisher policies or with permission. Permission for further reuse of this content should be sought from the publisher or the rights holder. This is the author created accepted manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at https://doi.org/10.1080/00927872.2019.1710161
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