Gorenstein dimensions and Hirsch length of groups

@misc{emmanouil2026gorenstein,
  title = {{Gorenstein dimensions and Hirsch length of groups}},
  author = {Ioannis Emmanouil and Konstantinos Golfis and Wei Ren},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.22386},
  eprint = {2608.22386},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.22386},
}

TY  - UNPB
TI  - Gorenstein dimensions and Hirsch length of groups
AU  - Ioannis Emmanouil
AU  - Konstantinos Golfis
AU  - Wei Ren
PY  - 2026
DA  - 2026/09/08
PB  - arXiv
DO  - 10.48550/arxiv.2608.22386
UR  - https://arxiv.org/abs/2608.22386
AB  - We study the relation between the Gorenstein homological and the Gorenstein cohomological dimension of groups. We prove that for every module of type FP_∞ over any ring the Gorenstein projective and the Gorenstein flat dimensions are always equal to each other. In particular, we have an equality Gcd_kG=Ghd_kG for every group G of type FP_∞ over a commutative coefficient ring k. For non-locally-finite groups in Kropholler's class scriptstyleLHF, we characterize the groups of Gorenstein (co)homological dimension one, in terms of actions on trees with finite stabilizers. The Hirsch length of a virtually soluble group G determines its Gorenstein (co)homological dimension, even when G has torsion: We show that Ghd_ZG =h(G) and, if G is countable, h(G)≤Gcd_ZG≤ h(G)+1. Some consequences for elementary amenable groups of finite Hirsch length are also obtained. Finally, we investigate the Gorenstein dimension of modules over certain algebras of groups with torsion and discuss applications to classifying spaces for proper actions.
ER  -