Year: 2025
Type: article
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}\llbracket G\rrbracket is a Sylvester domain. Moreover, the inner rank \mathrm{irk}_{\mathbb{F}_p\llbracket G\rrbracket}(A) of a matrix A over \mathbb{F}_{p}\llbracket G\rrbracket 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.
Source: Documenta Mathematica
DOI: 10.4171/dm/1034
Authors: Andrei Jaikin-Zapirain, Henrique Souza
Language: English
Topic: Rings, Modules, and Algebras
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W4414156334