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Henrique Souza

Publications and source records attributed to Henrique Souza.

5 recordsLinked to original sources
arXiv2026

Profinite detection of free products and free factors

Let $G$ be the fundamental group of a graph of finitely generated virtually free groups with virtually cyclic edge groups. We shaw that $G$ is cohomologically good if $G$ is residually finite. If $G$ is LERF, we prove that G splits non-trivially as a free product if and only if its profinite completion $\widehat{G}$ splits non-trivially as a free profinite product. Moreover, we are able to detect one-ended free factors of $G$ from $\widehat{G}$. As an application, we deduce that any profinitely rigid word in a finitely generated free group is universally profinitely rigid.

math.GR↗
arXiv2026

Sylvester domains and pro-$p$ groups

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 of $G$ over $\mathbb{F}_p$ is a Sylvester domain. Moreover the inner rank of a matrix $A$ over this completed group algebra 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 induced from $G \to G/G_i$, then the inner rank of $A$ equals $\lim_{i\to \infty} \frac{\operatorname{rk}_{\mathbb{F}_p} (A_i)}{|G:G_i|}$. 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.

math.GR↗
arXiv2024

Asymptotics of rational representations for algebraic groups

We study the asymptotic behaviour of the cohomology of subgroups $Γ$ 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 $Γ$ 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.

math.GR↗
arXiv2021

Inertia of retracts in Demushkin groups

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.

math.GR↗