Sylvester domains and pro-$p$ groups

1. Problem Statement

The paper addresses the structural properties of the completed group algebra $F_p[[G]]$ where $G$ is a finitely generated, torsion-free pro-$p$ group. Specifically, the authors investigate under what conditions $F_p[[G]]$ is a Sylvester domain.

A ring $R$ is a Sylvester domain if its inner rank function, $\text{irk}_R$, satisfies the axioms of a Sylvester matrix rank function. This property is significant because:

1. It implies the existence of a universal division ring of fractions $D_R$ into which $R$ embeds.
2. For any matrix $A$ over $R$, its rank over $D_R$ equals its inner rank $\text{irk}_R(A)$.
3. It relates to the pro-$p$ Atiyah conjecture, which predicts that the von Neumann Sylvester rank function $\text{rk}_G$ takes values in $p^{-n}\mathbb{Z}$ (or $\mathbb{Z}$ if $G$ is torsion-free).

The central question is: For which pro-$p$ groups $G$ is $F_p[[G]]$ a Sylvester domain, and does the inner rank coincide with the von Neumann rank $\text{rk}_G$?

The authors focus on free-by-$\mathbb{Z}_p$ pro-$p$ groups (groups $G$ containing an open normal subgroup $N$ that is free pro-$p$ with $G/N \cong \mathbb{Z}_p$) and their virtually free-by-$\mathbb{Z}_p$ counterparts.

2. Methodology

The authors employ a combination of combinatorial group theory, homological algebra, and non-commutative ring theory. Their approach involves several key methodological steps:

• Skew Power Series Decomposition:
  Unlike abstract free-by-cyclic groups where the group algebra is a crossed product $K[N] * \mathbb{Z}$, the pro-$p$ case does not admit such a decomposition. Instead, the authors represent $F_p[[G]]$ (where $G$ is free-by-$\mathbb{Z}_p$ with kernel $N$) as a skew power series ring:
  $$F_p[[G]] \cong F_p[[N]][[s; \sigma, \delta]]$$
  Here, $\sigma$ is the conjugation automorphism induced by a generator of $G/N$, and $\delta = \sigma - \text{id}$ is a derivation. They prove that $\delta$ is topologically nilpotent, ensuring the convergence of the multiplication in this ring.

• Mild Flag Presentations:
  To handle the structure of $F_p[[N]]$, the authors utilize the concept of mild pro-$p$ groups. They show that every finitely generated free-by-$\mathbb{Z}_p$ pro-$p$ group has a "mild flag presentation." This allows them to analyze the associated graded ring $\text{Gr}(F_p[[G]])$, proving it is a skew polynomial ring over a free algebra, which is a domain.

• Embedding into a Division Ring:
  The core of the proof involves constructing a chain of embeddings:
  $$F_p[[G]] \hookrightarrow \Lambda[[t]][[s; \sigma, \delta]] \hookrightarrow D[[t]][[s; \sigma, \delta]] \hookrightarrow Q$$
  where:

  • $\Lambda$ is a free associative algebra.
  • $D$ is the universal division ring of fractions of $\Lambda$.
  • $Q$ is the classical ring of fractions (Ore localization) of the Noetherian domain $D[[t]][[s; \sigma, \delta]]$.
    This establishes that $F_p[[G]]$ embeds into a division ring $Q$.

• Homological Criteria (Theorem 2.8):
  To prove $F_p[[G]]$ is a Sylvester domain, the authors apply a homological criterion from their previous work ([JZ20]). They must verify two conditions for the embedding $F_p[[G]] \to Q$:

  1. $\text{Tor}_1^{F_p[[G]]}(Q, Q) = 0$.
  2. Every finitely generated submodule of $Q$ is free (or has specific projective dimension properties).
     They achieve this by analyzing filtered modules and using the vanishing of Tor groups over the associated graded rings.

• Rank Approximation:
  To prove $\text{irk}_{F_p[[G]]} = \text{rk}_G$, they utilize a valuation $w$ on the division ring $Q$. They show that the induced rank function from the embedding is bounded by the von Neumann rank, and by analyzing the length of modules over the valuation ring, they establish equality.

3. Key Contributions and Results

Main Theorem (Theorem 1.1)

Let $G$ be a finitely generated, torsion-free pro-$p$ group containing an open free-by-$\mathbb{Z}_p$ pro-$p$ subgroup. Then:

1. $F_p[[G]]$ is a Sylvester domain.
2. The inner rank coincides with the von Neumann rank: $\text{irk}_{F_p[[G]]} = \text{rk}_G$.

Corollary 1.2 (Pro-$p$ Atiyah Conjecture)

Under the same conditions as Theorem 1.1, the rank function $\text{rk}_G$ takes only integer values. This provides new cases for the pro-$p$ Atiyah conjecture, extending beyond residually-$p$-adic analytic groups.

Structural Characterization

• Commensurability Invariance: The property of $F_p[[G]]$ being a Sylvester domain (and the equality of ranks) is a commensurability invariant. If it holds for an open subgroup, it holds for the whole group.
• Cohomological Dimension: If $F_p[[G]]$ is a Sylvester domain, then $G$ has cohomological dimension $\le 2$.
• p-adic Analytic Groups: The authors characterize $p$-adic analytic groups for which $F_p[[G]]$ is a Sylvester domain: they are precisely those isomorphic to $\mathbb{Z}_p$ or $\mathbb{Z}_p \rtimes \mathbb{Z}_p$.

Lück Approximation (Section 8)

The paper applies these results to abstract finitely generated subgroups $\Gamma$ of free-by-$\mathbb{Z}_p$ pro-$p$ groups. They prove a specific case of the Lück approximation conjecture in positive characteristic ($p$):
$$\lim_{i \to \infty} \text{rk}_{\Gamma/H_i}(A) = \text{rk}_{F_p[\Gamma]}(A)$$
where $H_i$ is a chain of finite index normal subgroups. This confirms that the rank of matrices over the abstract group algebra can be approximated by ranks over finite quotients.

4. Significance

1. Resolution of a Major Conjecture Case: The paper confirms the pro-$p$ Atiyah conjecture (specifically the integrality of ranks) for a broad and important class of pro-$p$ groups (virtually free-by-$\mathbb{Z}_p$), which includes infinite Demushkin groups.
2. Bridging Abstract and Pro-$p$ Theory: It demonstrates that while proofs in combinatorial pro-$p$ group theory often differ from their abstract counterparts (e.g., the lack of crossed product decomposition), the resulting structural properties (Sylvester domain status) can be analogous or even stronger.
3. New Tools for Non-Commutative Rings: The construction of the embedding into the division ring $Q$ via skew power series rings and the use of "mild flag presentations" provide a robust framework for studying completed group algebras of more complex pro-$p$ groups.
4. Lück Approximation in Positive Characteristic: Prior to this work, Lück approximation in characteristic $p$ was largely unknown outside of amenable groups. This result extends the validity of the approximation to non-amenable, virtually free-by-$\mathbb{Z}_p$ groups, a significant advancement in the field.

5. Conclusion

Jaikin-Zapirain and Souza successfully establish that the completed group algebras of virtually free-by-$\mathbb{Z}_p$ pro-$p$ groups are Sylvester domains. By constructing a universal division ring of fractions and proving the equality of inner and von Neumann ranks, they validate the pro-$p$ Atiyah conjecture for these groups and provide a powerful tool for calculating ranks via finite approximations. This work deepens the understanding of the algebraic structure of pro-$p$ groups and their representations.