Beauville structures in finite p-groups

We study the existence of (unmixed) Beauville structures in finite $p$-groups, where $p$ is a prime. First of all, we extend Catanese's characterisation of abelian Beauville groups to finite $p$-groups satisfying certain conditions which are much weaker than commutativity. This result applies to all known families of $p$-groups with a good behaviour with respect to powers: regular $p$-groups, powerful $p$-groups and more generally potent $p$-groups, and (generalised) $p$-central $p$-groups. In particular, our characterisation holds for all $p$-groups of order at most $p^p$, which allows us to determine the exact number of Beauville groups of order $p^5$, for $p\ge 5$, and of order $p^6$, for $p\ge 7$. On the other hand, we determine which quotients of the Nottingham group over $\mathbb{F}_p$ are Beauville groups, for an odd prime $p$. As a consequence, we give the first explicit infinite family of Beauville $3$-groups, and we show that there are Beauville $3$-groups of order $3^n$ for every $n\ge 5$

Outer commutator words are uniformly concise

We prove that outer commutator words are uniformly concise, i.e. if an outer commutator word w takes m different values in a group G, then the order of the verbal subgroup w(G) is bounded by a function depending only on m and not on w or G. This is obtained as a consequence of a structure theorem for the subgroup w(G), which is valid if G is soluble, and without assuming that w takes finitely many values in G. More precisely, there is an abelian series of w(G), such that every section of the series can be generated by values of w all of whose powers are also values of w in that section. For the proof of this latter result, we introduce a new representation of outer commutator words by means of binary trees, and we use the structure of the trees to set up an appropriate induction

On the congruence subgroup property for GGS-groups

We show that all GGS-groups with non-constant defining vector satisfy the congruence subgroup property. This provides, for every odd prime $p$, many examples of finitely generated, residually finite, non-torsion groups whose profinite completion is a pro-$p$ group, and among them we find torsion-free groups. This answers a question of Barnea. On the other hand, we prove that the GGS-group with constant defining vector has an infinite congruence kernel and is not a branch group.Comment: v2 incorporates referee suggestions (final version

Hausdorff dimension in RR-analytic profinite groups

We study the Hausdorff dimension of R-analytic subgroups in an R-analytic profinite group, where R is a pro-p ring whose asso- ciated graded ring is an integral domain. In particular, we prove that the set of such Hausdorff dimensions is a finite subset of the rational numbers.Comment: 7 page

A restriction on centralizers in finite groups

For a given m>=1, we consider the finite non-abelian groups G for which |C_G(g):|<=m for every g in G\Z(G). We show that the order of G can be bounded in terms of m and the largest prime divisor of the order of G. Our approach relies on dealing first with the case where G is a non-abelian finite p-group. In that situation, if we take m=p^k to be a power of p, we show that |G|<=p^{2k+2} with the only exception of Q_8. This bound is best possible, and implies that the order of G can be bounded by a function of m alone in the case of nilpotent groups

Telecom submarine cables, a business compatible with the environment protection

All playing sgents in marine operation; telecoms, cable manufactures and cableship operators, are more and more involved in making their increasingly watched and regulated activities to be compatible with sustainable and live oceans.Peer Reviewe