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This is a Quantitative Study study.

September 12, 2025Open Access

Sylvester domains and pro-p groups

Key Points

The completed group algebra is identified as a Sylvester domain, which has significant algebraic implications.
The inner rank of a matrix A can be effectively calculated using finite quotients of the group G.
This work presents a specific case of the mod p Lück approximation relevant to free-by-Zp pro-p groups.
The results provide insights into the algebraic structure of finitely generated torsion-free pro-p groups.

Abstract

Let G be a finitely generated torsion-free pro- p group containing an open free-by- Z₏ pro- p subgroup. We show that the completed group algebra F₏ G is a Sylvester domain. Moreover, the inner rank irk₅䂹 ₆ (A) of a matrix A over F₏ 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- Z₏ pro- p groups.

Sylvester domains and pro-p groups

Key Points

Abstract

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