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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 enough, the model produces emph{infinite} $n$-periodic groups. As an application, we obtain, for every fixed large enough $n$, for every $pin (1,infty)$ an infinite $n$-periodic group with fixed points for all isometric actions on $L^p$-spaces. Our main contribution is to show that certain random triangular groups are uniformly acylindrically hyperbolic.
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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 be used only for density < 1/5. The strategy employed might be potentially extended in future to all densities < 1/4.
We show that for a fixed k, Gromov random groups with any positive density have no non-trivial degree-k representations over any field, a.a.s. This is especially interesting in light of the results of Agol, Ollivier and Wise that when the density is less than 1/6 such groups have a faithful linear representation over the rationals, a.a.s.
We prove that a random group in the triangular density model has, for density larger than 1/3, fixed point properties for actions on $L^p$-spaces (affine isometric, and more generally $(2-2epsilon)^{1/2p}$-uniformly Lipschitz) with $p$ varying in an interval increasing with the set of generators. In the same model, we establish a double inequality between the maximal $p$ for which $L^p$-fixed point properties hold and the conformal dimension of the boundary. In the Gromov density model, we prove that for every $p_0 in [2, infty)$ for a sufficiently large number of generators and for any density larger than 1/3, a random group satisfies the fixed point property for affine actions on $L^p$-spaces that are $(2-2epsilon)^{1/2p}$-uniformly Lipschitz, and this for every $pin [2,p_0]$. To accomplish these goals we find new bounds on the first eigenvalue of the p-Laplacian on random graphs, using methods adapted from Kahn and Szemeredis approach to the 2-Laplacian. These in turn lead to fixed point properties using arguments of Bourdon and Gromov, which extend to $L^p$-spaces previous results for Kazhdans Property (T) established by Zuk and Ballmann-Swiatkowski.
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 used by T. C. Burness, M. W. Liebeck and A. Shalev, it was asked by L. Pyber that for coprime linear groups $Gleq GL(V)$, whether there exists a constant $c$ such that the probability of that a random $c$-tuple is a base for $G$ tends to 1 as $|V|toinfty$. While the answer to this question is negative in general, it is positive under the additional assumption that $G$ is even primitive as a linear group. In this paper, we show that almost all $11$-tuples are bases for coprime primitive linear groups.
We prove that all invariant random subgroups of the lamplighter group $L$ are co-sofic. It follows that $L$ is permutation stable, providing an example of an infinitely presented such a group. Our proof applies more generally to all permutational wreath products of finitely generated abelian groups. We rely on the pointwise ergodic theorem for amenable groups.
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