Finite generation of iterated wreath products in product action

Let $\mathcal{S}$ be a sequence of finite perfect transitive permutation groups with uniformly bounded number of generators. We prove that the infinitely iterated wreath product in product action of the groups in $\mathcal{S}$ is topologically finitely generated, provided that the actions of the groups in $\mathcal{S}$ are not regular. We prove that our bound has the right asymptotic behaviour. We also deduce that other infinitely iterated mixed wreath products of groups in $\mathcal{S}$ are finitely generated. Finally we apply our methods to find explicitly two generators of infinitely iterated wreath products in product action of special sequences $\mathcal{S}$.Comment: 9 pages. Updated version with some added references. arXiv admin note: substantial text overlap with arXiv:1412.780

Hereditarily just infinite profinite groups with complete Hausdorff dimension spectrum

We prove that the inverse limit of certain iterated wreath products in product action have complete Hausdorff dimension spectrum with respect to their unique maximal filtration of open normal subgroups. Moreover we can produce explicitly subgroups with a specified Hausdorff dimension.Comment: 8 pages, includes minor correction

Commuting and product-zero probability in finite rings

Let cp(R) be the probability that two random elements of a finite ring R commute and zp(R) the probability that the product of two random elements in R is zero. We show that if cp(R)=e, then there exists a Lie-ideal D in the Lie-ring (R,[.,.]) with e-bounded index and with [D,D] of e-bounded order. If zp(R)=e, then there exists an ideal D in R with e-bounded index and D^2 of e-bounded order. These results are analogous to the well-known theorem of P. Neumann on the commuting probability in finite groups