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 grouptheoretic 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 (We include an exposition of Kozma's previously unpublished argument, with slightly improved results, for completeness.)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 language  English 

Pages (fromto)  9851005 
Number of pages  21 
Journal  Groups, Geometry and Dynamics 
Volume  10 
Issue number  3 
DOIs  
Publication status  Published  2016 
Keywords
 Random groups
 density
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Dr John M Mackay
 School of Mathematics  Associate Professor in Pure Mathematics
 Probability, Analysis and Dynamics
 Pure Mathematics
 Analysis
Person: Academic , Member