Research | Henrique A. Souza

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List of publications

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

Abstract: 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 $SL_2$. 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 quadratic case.

Sylvester domains and pro-$p$ groups (2024). Joint with Andrei Jaikin-Zapirain. Submitted. On ArXiv: https://arxiv.org/abs/2402.14130

Abstract: 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 $\operatorname{irk}_{\mathbb{F}_p[\![G]\!]}(A)$ of a matrix $A$ over $\mathbb{F}_p[\![G]\!]$ can be calculated by approximation by ranks corresponding to finite quotients of $G$, that is, if $G=G_1>G_2>\ldots$ is a chain of normal open subgroups of $G$ with trivial intersection and $A_i$ is the matrix over $\mathbb{F}_p[G/G_i]$ obtained from the matrix $A$ by applying the natural homomorphism $\mathbb{F}_p[\![G]\!]\to \mathbb{F}_p[G/G_i]$, then
$\operatorname{irk}_{\mathbb{F}_p[\![G]\!]}(A)=\displaystyle \lim_{i\to \infty} \frac{\operatorname{rk}_{\mathbb{F}_p} (A_i)}{|G:G_i|}$. 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 in 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

Abstract: Exploring inequalities regarding the rank and relation gradients of pro-p modules and building upon recent results of Y. Antolín, A. Jaikin-Zapiran 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 absence of a virtual decomposition 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 $H^1(G,\mathbb{F}_p)$ is finite, $H^2(G,\mathbb{F}_p)$ has dimension 1 and the cup product $H^1(G,\mathbb{F}_p) \times H^1(G,\mathbb{F}_p) \to H^2(G,\mathbb{F}_p)$ 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 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.