密度为一半的随机组的阈值更高
A sharper threshold for random groups at density one-half
摘要：
在随机群的密度模型中，我们考虑具有任意固定数量m的生成元和许多长度为l的随机关系子的表示，将l发送到无穷大。如果d是一个“密度”参数，用来衡量关联子数目与关联子长度相比的指数增长性，那么许多群论性质在不同的d值下一般为真或一般为假。这个密度模型的签名定理是从平凡到双曲的相变：对于d < 1/2，随机群是a.a.s.。无限双曲，而对于d > 1/2，随机群是a.a.s.点一两个。我们研究密度阈值d = 1/2的随机组。Kozma发现，平凡的群体是通用的范围内的增长率在d = 1/2;我们表明，无限双曲群是通用的在不同的范围。(We包括对Kozma以前未发表论证的阐述，为了完整起见，结果略有改进。）
英文摘要：
In the density model of random groups, we consider presentations with any fixed number m of generators and many random relators of length l, sending l to infinity. 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 (We include an exposition of Kozma's previously unpublished argument, with slightly improved results, for completeness.)
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