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A sharper threshold for random groups at density one-half. (English) Zbl 1394.20044
Summary: In the theory of random groups, we consider presentations with any fixed number $$m$$ of generators and many random relators of length $$\ell$$, sending $$\ell \rightarrow \infty$$. 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 (We include an exposition of Kozma’s previously unpublished argument, with slightly improved results, for completeness.)

##### MSC:
 20P05 Probabilistic methods in group theory 20F06 Cancellation theory of groups; application of van Kampen diagrams 20F67 Hyperbolic groups and nonpositively curved groups
##### Keywords:
random groups; density
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