Gorenstein versions of type $Φ$ groups and some Gcd-cd, Ghd-hd coincidences

TL;DR

This work develops Gorenstein analogues of Talelli's type $Φ$ groups and establishes characterizations via finiteness of Gorenstein (co)dimensions, aligning them with classical invariants over a base ring $A$ under cohomological finiteness hypotheses. It extends Kropholler's LH hierarchy by substituting the finite subgroup base with the class of type $Φ$ groups and introduces action on AG-modules, characteristic modules, and related dimensions. The main results show that, for torsion-free groups in the LH class, Gorenstein and classical cohomological dimensions coincide when certain finiteness conditions are met, and similar equalities hold for Gorenstein and classical homological dimensions. These findings generalize Talelli's results to broader ring settings and provide a unified framework linking type $Φ$ phenomena, Gorenstein homological algebra, and higher-dimensional classifying spaces.

Abstract

In this short note, we characterise some Gorenstein versions of the concept of a group being of type $Φ$ as introduced by Olympia Talelli. And, we also generalize a different Talelli result regarding the coincidence of the classical and the Gorenstein cohomological dimension of torsion-free groups in Kropholler's $\LH\mathscr{F}$ class.

Theorems & Definitions (11)

• Definition 2.1
• Theorem 3.1
• Corollary 3.2
• proof
• Theorem 3.3
• proof
• Corollary 3.4
• Remark 3.5
• Lemma 3.6
• Theorem 3.7