Jon González-Sánchez

Analytic pro-p groups of small dimensions

Abstract According to Lazard, every p-adic Lie group contains an open pro-p subgroup which is saturable. This can be regarded as the starting point of p-adic Lie theory, as one can naturally associate to every saturable pro-p group G a Lie lattice L(G) over the p-adic integers. Essential features of saturable pro-p groups include that they are torsion-free and p-adic analytic. In the present paper we prove a converse result in small dimensions: every torsion-free p-adic analytic pro-p group of dimension less than p is saturable. This leads to useful consequences and interesting questions. For instance, we give an effective classification of 3-dimensional soluble torsion-free p-adic analytic pro-p groups for p > 3. Our approach via Lie theory is comparable with the use of Lazards correspondence in the classification of finite p-groups of small order.

Kirillov's Orbit Method for p-Groups and Pro-p Groups

In this text, we study Kirillovs orbit method in the context of Lazards p-saturable groups when p is an odd prime. Using this approach we prove that the orbit method works in the following cases: torsion free p-adic analytic pro-p groups of dimension smaller than p, pro-p Sylow subgroups of classical groups over ℤ p of small dimension and for certain families of finite p-groups.

The representation zeta function of a FAb compact p-adic Lie group vanishes at -2

Let G by compact p-adic Lie group and suppose that G is FAb, i.e., that H/(H,H) is finite for every open subgroup H of G. The representation zeta functionG(s) = P �2Irr(G) �(1) s encodes the distribution of continuous irreducible complex characters of G. For p � 3 it is known thatG(s) defines a meromorphic function on C. Wedderburns structure theorem for semisimple algebras implies that �G( 2) = |G| for finite G. We complement this classic result by proving thatG( 2) = 0 for infinite G, assuming p � 3.

Analyzing group based matrix multiplication algorithms

We review different group based algorithms for matrix multiplication and discuss the relations between the combinatorial properties of the used group and the complexity of these algorithms. We introduce a variant of an algorithm based on the ideas exposed in [4] well-adapted for experimentation. Finally we show how this approach can also be used for matrix multiplication over a field with characteristic different from 2.

Hausdorff dimension in R-analytic profinite groups

Abstract We study the Hausdorff dimension of R-analytic subgroups in an infinite R-analytic profinite group, where R is a pro-p ring whose associated 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.

Cohomology of p-groups of nilpotency class smaller than p

Abstract Let p be a prime number, let d be an integer and let G be a d-generated finite p-group of nilpotency class smaller than p. Then the number of possible isomorphism types for the mod p cohomology algebra H * ⁢ ( G ; 𝔽 p ) {H^{*}(G;{\mathbb{F}}_{p})} is bounded in terms of p and d.

A cohomological criterion for p-solvability

Let G be a finite group, let p be a prime, and let P be a Sylow p-subgroup of G. In this note we give a cohomological criterion for the p-solvability of G depending on the cohomology in degree 1 with coefficients in

Omega subgroups of pro-p groups

\mathbb F_p