Research Profile

Profinite groups, rank functions, and algebraic structures

Henrique Souza

Research map

Early independence: Demushkin groups

Solo paper, J. Group Theory, 2022, 11 p.

  • Retracts of Demushkin groups
  • Pro-p analogue of phenomena for surface groups
  • New methods: the surface proof does not carry over.

Main pillar I: completed group algebras

With A. Jaikin-Zapirain, Documenta Math. 2025, 50 p.

Ring-theoretic properties of \mathbb F_p[\![G]\!] for G virtually free-by-\mathbb Z_p.

Main themes:

  • Sylvester rank functions
  • Noncommutative localization
  • Filtered/graded methods
  • Approximation by finite quotients

My contribution to this line

  • Structural analysis of virtually free-by-\mathbb Z_p pro-p groups
  • Topological viewpoint on localization of profinite rings
  • Development of rank functions as a core technical tool

Thesis synthesis: examples, conjectures, and future directions beyond the paper.

Main pillar II: asymptotics of rational representations

With L. Guerrero-Sánchez, preprint, 33 p.

Reinterprets the asymptotic behaviour of rational representations in terms of rank functions on group algebras.

Motivation:

  • twisted cohomology
  • approximation of \ell^2-Betti numbers
  • non-arithmetic lattices in \mathrm{SL}_2(\mathbb C)
  • higher-rank conjectures

Broader pro-p and profinite program

Further preprints:

  • With P. Zalesskii: \mathrm{PD}^2 groups and pro-p limit group analogues
  • With P. Zalesskii and A. Jaikin-Zapirain: profinite rigidity of graphs of virtually free groups

Main technical themes:

  • Pro-p analogues of simple closed curves
  • Translating profinite splitting data into discrete splitting data via derivations

Current and future work

In preparation with A. Jaikin-Zapirain and N. Otmen:

  • profinite \mathrm{PD}^3 groups
  • non-large and non-analytic examples
  • exponential torsion homology growth

Future direction:

  • asymptotic cohomology of higher-rank lattices through completed universal enveloping algebras (w/ K. Ardakov)

Fit and summary

My profile sits between:

  • profinite group theory
  • completed group rings and algebras
  • cohomological methods
  • representation theory

At UnB, I would contribute through:

  • research at the intersection of groups and algebras
  • international collaborations
  • student training in profinite methods, rank functions, and noncommutative algebra