Sylvester domains and pro-$p$ groups

TL;DR

The paper proves that for finitely generated, torsion-free pro-$p$ groups $G$ with an open free-by-$\mathbb{Z}_p$ subgroup, the completed group algebra $\mathbb{F}_p[[G]]$ is a Sylvester domain and the inner rank equals the pro-$p$-rank $\operatorname{rk}_G$, enabling a mod-$p$ Lück-type approximation. The authors develop a framework using skew power series rings, universal division rings, and homological finiteness to embed $\mathbb{F}_p[[G]]$ into a division ring and to establish Tor-vanishing and graded-depth properties that transfer to $S=\mathbb{F}_p[[G]]$. The key technical advance is the introduction of mild flag pro-$p$ groups, whose Graded rings are domains, with valuations that render the completed group algebra a Sylvester domain; this yields that $\operatorname{rk}_G$ takes integer values and supports a pro-$p$ Atiyah-type conjecture in this setting. Consequently, the main theorem implies a robust Lück approximation for abstract finitely generated subgroups of free-by-$\mathbb{Z}_p$ pro-$p$ groups, extending positive-characteristic rank theory to new non-abelian pro-$p$ contexts and providing tools for further exploration of rank functions in profinite settings.

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 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.

Theorems & Definitions (105)

• Theorem 1.1
• Corollary 1.2
• Proposition 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• proof : Proof of Proposition \ref{['prop-exmp-not-free-by-zp']}
• Lemma 2.4
• proof