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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 (We include an exposition of Kozmas previously unpublished argument, with slightly improved results, for completeness.)
We study a random group G in the Gromov density model and its Cayley complex X. For density < 5/24 we define walls in X that give rise to a nontrivial action of G on a CAT(0) cube complex. This extends a result of Ollivier and Wise, whose walls could
The minimal base size $b(G)$ for a permutation group $G$, is a widely studied topic in the permutation group theory. Z. Halasi and K. Podoski proved that $b(G)leq 2$ for coprime linear groups. Motivated by this result and the probabilistic method use
We prove a freeness theorem for low-rank subgroups of one-relator groups. Let $F$ be a free group, and let $win F$ be a non-primitive element. The primitivity rank of $w$, $pi(w)$, is the smallest rank of a subgroup of $F$ containing $w$ as an imprim
We introduce a model for random groups in varieties of $n$-periodic groups as $n$-periodic quotients of triangular random groups. We show that for an explicit $d_{mathrm{crit}}in(1/3,1/2)$, for densities $din(1/3,d_{mathrm{crit}})$ and for $n$ large
In this paper, we prove that two-generator one-relator groups with depth less than or equal to 3 can be effectively embedded into a tower of HNN-extensions in which each group has the effective standard normal form. We give an example to show how to