Homological growth of Artin kernels in positive characteristic

TL;DR

This work extends Lück's Approximation Theorem to positive characteristic for RAAGs, Bestvina–Brady groups, and related residually finite rationally soluble groups by deploying agrarian invariants via Hughes–free division rings. It proves that mod $p$ homology growth equals the dimension of agrarian homology, independently of the residual chain, and provides a universal lower bound comparing agrarian Betti numbers with homology gradients. The authors develop a Mayer–Vietoris–type spectral sequence for spaces with confident covers to compute explicit invariants for Artin kernels, graph products, Artin groups, and hyperplane arrangement complements, yielding precise formulas in terms of nerve and link data. They then apply these computations to fibring, amenable category, and minimal volume entropy, deriving criteria for fibring behavior and entropy, and establishing connections between algebraic and geometric invariants in these residually finite groups.

Abstract

We prove an analogue of the Lück Approximation Theorem in positive characteristic for certain residually finite rationally soluble (RFRS) groups including right-angled Artin groups and Bestvina--Brady groups. Specifically, we prove that the mod $p$ homology growth equals the dimension of the group homology with coefficients in a certain universal division ring and this is independent of the choice of residual chain. For general RFRS groups we obtain an inequality between the invariants. We also consider a number of applications to fibring, amenable category, and minimal volume entropy.

Theorems & Definitions (68)

• Theorem 1.1: Lück, LuckApprox
• Conjecture A
• Theorem B
• Theorem C
• Theorem D
• Theorem 1.2
• Theorem 2.1: Linnell LinnellDivRings93
• Theorem 2.2: Jaikin-Zapirain JaikinZapirain2020THEUO
• Lemma 2.3: DavisOkun2012
• Remark 3.1