A sharper threshold for random groups at density one-half

Moon Duchin, Kasia Jankiewicz, Shelby Klimer, Samuel Lelièvre, John Mackay, Andrew Sánchez

Research output: Contribution to journalArticle (Academic Journal)

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Abstract

In the theory of random groups, we consider presentations with any fixed number mm of generators and many random relators of length ℓℓ, sending ℓ→∞ℓ→∞. If dd is a „density“ parameter measuring the rate of exponential growth of the number of relators compared to the length of relators, then many group-theoretic properties become generically true or generically false at different values of dd. The signature theorem for this density model is a phase transition from triviality to hyperbolicity: for d<1/2d<1/2, random groups are a.a.s. infinite hyperbolic, while for d>1/2d>1/2, random groups are a.a.s. order one or two. We study random groups at the density threshold d=1/2d=1/2. Kozma had found that trivial groups are generic for a range of growth rates at d=1/2d=1/2; we show that infinite hyperbolic groups are generic in a different range. (We include an exposition of Kozma's previously unpublished argument, with slightly improved results, for completeness.)
Original languageEnglish
Pages (from-to)985-1005
Number of pages21
JournalGroups, Geometry and Dynamics
Volume10
Issue number3
DOIs
Publication statusPublished - 2016

Keywords

  • Random groups
  • density

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    Duchin, M., Jankiewicz, K., Klimer, S., Lelièvre, S., Mackay, J., & Sánchez, A. (2016). A sharper threshold for random groups at density one-half. Groups, Geometry and Dynamics, 10(3), 985-1005. https://doi.org/10.4171/GGD/374