Geodesic
A sharper threshold for random groups at density one-half
Moon Duchin1 ; Kasia Jankiewicz2 orcid ; Shelby C. Kilmer3 ; Samuel Lelièvre4 ; John M. Mackay5 ; Andrew P. Sánchez1
1Tufts University, Medford, USA
2McGill University, Montreal, Canada
3University of Utah, Salt Lake City, USA
4Université Paris-Sud, Orsay, France
5University of Bristol, UK
Groups, geometry, and dynamics, Tome 10 (2016) no. 3, pp. 985-1005

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DOI

In the theory of random groups, we consider presentations with any fixed number m of generators and many random relators of length l, sending l→∞. If d 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 d. The signature theorem for this density model is a phase transition from triviality to hyperbolicity: for d<1/2, random groups are a.a.s. infinite hyperbolic, while for d>1/2, random groups are a.a.s. order one or two. We study random groups at the density threshold d=1/2. Kozma had found that trivial groups are generic for a range of growth rates at d=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 (We include an exposition of Kozma's previously unpublished argument, with slightly improved results, for completeness.)
DOI : 10.4171/ggd/374
Classification : 20-XX
Mots-clés : Random groups, density

Moon Duchin  1   ; Kasia Jankiewicz  2   ; Shelby C. Kilmer  3   ; Samuel Lelièvre  4   ; John M. Mackay  5   ; Andrew P. Sánchez  1

1 Tufts University, Medford, USA
2 McGill University, Montreal, Canada
3 University of Utah, Salt Lake City, USA
4 Université Paris-Sud, Orsay, France
5 University of Bristol, UK 
Moon Duchin; Kasia Jankiewicz; Shelby C. Kilmer; Samuel Lelièvre; John M. Mackay; Andrew P. Sánchez. A sharper threshold for random groups at density one-half. Groups, geometry, and dynamics, Tome 10 (2016) no. 3, pp. 985-1005. doi: 10.4171/ggd/374
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     title = {A sharper threshold for random groups at density one-half},
     journal = {Groups, geometry, and dynamics},
     pages = {985--1005},
     year = {2016},
     volume = {10},
     number = {3},
     doi = {10.4171/ggd/374},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/ggd/374/}
}
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