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List of publications

From newest to oldest

Asymptotics of rational representations for algebraic groups (2024). Joint with Lander Guerrero Sánchez. Submitted. On Arxiv: https://arxiv.org/abs/2405.17360

We study the asymptotic behaviour of the cohomology of subgroups \(\Gamma\) of an algebraic group \(G\) with coefficients in the various irreducible rational representations of \(G\) and raise a conjecture about it. Namely, we expect that the dimensions of these cohomology groups approximate the \(\ell^2\)-Betti numbers of \(\Gamma\) with a controlled error term. We provide positive answers when \(G\) is a product of copies of \(\mathrm{SL}_2\). As an application, we obtain new proofs of J. Lott’s and W. Lück’s computation of the \(\ell^2\)-Betti numbers of hyperbolic 3-manifolds and W. Fu’s upper bound on the growth of cusp forms for non totally real fields, which is sharp in the imaginary quadractic case.

Sylvester domains and pro-\(p\) groups (2024). Joint with Andrei Jaikin-Zapirain. In Documenta Mathematica (Online first). https://doi.org/10.4171/dm/1034. Also on ArXiv: https://arxiv.org/abs/2402.14130

Let \(G\) be a finitely generated torsion-free pro-\(p\) group containing an open free-by-\(\mathbb{Z}_p\) pro-\(p\) subgroup. We show that the completed group algebra \(\mathbb{F}_p[\![G]\!]\) is a Sylvester domain. Moreover the inner rank \(\mathrm{irk}_{\mathbb{F}_p[\![G]\!]}(A)\) of a matrix \(A\) over \(\mathbb{F}_p[\![G]\!]\) can be calculated by approximation by ranks corresponding to finite quotients of \(G\). As a consequence, we obtain a particular case of the mod \(p\) Lück approximation for abstract finitely generated subgroups of free-by-\(\mathbb{Z}_p\) pro-\(p\) groups.

Inertia of retracts of Demushkin groups (2022). In Journal of Group Theory (Vol. 26, Issue 3). https://doi.org/10.1515/jgth-2022-0055. Also on ArXiv: https://arxiv.org/abs/2111.03060.

Exploring inequalities regarding the rank and relation gradients of pro-\(p\) modules and building upon recent results of Y. Antolín, A. Jaikin-Zapirain and M. Shusterman, we prove that every retract of a Demushkin group is inert in the sense of the Dicks-Ventura Inertia Conjecture.

M.Sc. Thesis

Defense slides

My thesis, presented under the advice of Theo Zapata, explored the combinatorial properties of Dëmushkin groups. Its goal was to be a guide through recent results about Dëmushkin groups concerning the finiteness properties of subgroups and virtual retracts. More precisely, pro-\(p\) analogues of Howson’s theorem, the Hanna Neumann inequality, the presence of the virtual retractions property and the abscence of virtual decompositions as a free pro-\(p\) product.

During my master degree, my research was focused on a particular class of topological groups called Dëmushkin groups. A Dëmushkin group is a pro-\(p\) group that satisfies Poincaré duality in dimension 2, that is, \(G\) is a Dëmushkin group if \(\mathrm{H}^1(G,\mathbb{F}_p)\) is finite, \(\mathrm{H}^2(G,\mathbb{F}_p)\) has dimension 1 and the cup product \(\mathrm{H}^1(G,\mathbb{F}_p) \times \mathrm{H}^1(G,\mathbb{F}_p) \to \mathrm{H}^2(G,\mathbb{F}_p)\) is a non-degenerate bilinear form. Dëmushkin groups appear naturally as Galois groups of maximal \(p\)-extensions of local fields, as pro-\(p\) completions of surface groups and as maximal pro-\(p\) quotients of étale fundamental groups of smooth projective curves.

Dëmushkin groups have lots of great properties. They are topologically finitely generated and possess a topological presentation given by a single well known relation. They have strong hereditary, finiteness and virtual decomposition properties similar to those found in free pro-\(p\) groups. There is a lot of research about them, and many important developments have been made in the past decades.