Related papers: Beauville structures in $p$-central quotients

A Beauville surface is a rigid complex surface of general type, isogenous to a higher product by the free action of a finite group $G$, called a Beauville group. In \cite{GT}, Gonz\'alez-Diez and Torres-Teigell find the number of…

In this note, we study rigid complex manifolds that are realized as quotients of a product of curves by a free action of a finite group. They serve as higher-dimensional analogues of Beauville surfaces. Using uniformization, we outline the…

A finitely generated virtually free pro-p group with finite centralizers of its torsion elements is the free pro-p product of finite p-groups and a free pro-p factor.

We prove that the word problem of the Brin-Thompson group nV over a finite generating set is coNP-complete for every n \ge 2. It is known that the groups nV are an infinite family of infinite, finitely presented, simple groups. We also…

We consider the four structures $(\mathbb{Z}; \mathrm{Sqf}^\mathbb{Z})$, $(\mathbb{Z}; <, \mathrm{Sqf}^\mathbb{Z})$, $(\mathbb{Q}; \mathrm{Sqf}^\mathbb{Q})$, and $(\mathbb{Q}; <, \mathrm{Sqf}^\mathbb{Q})$ where $\mathbb{Z}$ is the additive…

For a prime number $p$ and a free profinite group $S$, let $S^{(n,p)}$ be the $n$th term of its lower $p$-central filtration, and $S^{[n,p]}$ the corresponding quotient. Using tools from the combinatorics of words, we construct a canonical…

A Beauville surface is a rigid complex surface of general type, isogenous to a higher product by the free action of a finite group, called a Beauville group. Here we consider which characteristically simple groups can be Beauville groups.…

Extending results of Bauer, Catanese and Grunewald, and of Fuertes and Gonz\'alez-Diez, we show that Beauville surfaces of unmixed type can be obtained from the groups L_2(q) and SL_2(q) for all prime powers q>5, and the Suzuki groups…

Let G be the free product of groups A and B with commuting subgroups H \leqslant A and K \leqslant B, and let C be the class of all finite groups or the class of all finite p-groups. We derive the description of all C-separable cyclic…

Let \Gamma be a finitely presentable pro-p group with a nontrivial finitely generated closed normal subgroup N of infinite index. Then def(\Gamma)\leq 1, and if def(\Gamma)=1 then \Gamma is a pro-p duality group of dimension 2, N is a free…

A conjecture of Mordell states that if $p$ is a prime and $p$ is congruent to $3$ mod $4$, then $p$ does not divide $y$ where $(x,y)$ is the fundamental solution to $x^{2}-py^{2}=1$. The conjecture has been verified for primes not exceeding…

A B-group is a group such that all its minimal generating sets (with respect to inclusion) have the same size. We prove that the class of finite B-groups is closed under taking quotients and that every finite B-group is solvable. Via a…

We classify all groups of order $p^5$ with non-trivial unramified Brauer groups. We show that if $p>3$, then there are precisely $\gcd (p-1,4)+\gcd (p-1,3)+1$ such groups.

We prove that every free metabelian non--cyclic group has a finitely generated isolated subgroup which is not separable in the class of nilpotent groups. As a corollary we prove that for every prime number $p$ an arbitrary free metabelian…

A long-standing conjecture asserts that every finite nonabelian $p$-group has a non-inner automorphism of order $p$. In this paper we prove the conjecture for finite $p$-groups of coclass $4$ and coclass $5$ ($p\ge 5$). We also prove the…

In this paper we consider Chevalley groups over commutative rings with~$1$, constructed by irreducible root systems of rank $>1$. We always suppose that for the systems $A_2, B_\ell, C_\ell, F_4, G_2$ our rings contain $1/2$ and for the…

We discuss Beauville groups whose corresponding Beauville surfaces are either always strongly real or never strongly real producing several infinite families of examples.

Let G be a finitely generated infinite pro-p group acting on a pro-p tree such that the restriction of the action to some open subgroup is free. Then we prove that G splits as a pro-p amalgamated product or as a pro-p HNN-extension over an…

We prove the Farrell-Jones conjecture for free-by-cyclic groups. The proof uses recently developed geometric methods for establishing the Farrell-Jones Conjecture.

The existence of an infinite simple boundedly generated 2-generated group and the existence of a boundedly simple 2-generated group containing a free non-cyclic subgroup are proved.