Let π be a group equipped with an action of a second group G by automorphisms. We define the equivariant cohomological dimension cdG(π), the equivariant geometric dimension catG(π), and the equivariant Lusternik–Schnirelmann category gdG(π) in terms of the Bredon dimensions and classifying space of the family of subgroups of the semi-direct product π⋊G consisting of sub-conjugates of G. When G is finite, we extend theorems of Eilenberg–Ganea and Stallings–Swan to the equivariant setting, thereby showing that all three invariants coincide (except for the possibility of a G-group π with catG(π)=cdG(π)=2 and gdG(π)=3). A main ingredient is the purely algebraic result that the cohomological dimension of any finite group with respect to any family of proper subgroups is greater than one. This implies a Stallings–Swan type result for families of subgroups which do not contain all finite subgroups.
Mark Grant; Ehud Meir; Irakli Patchkoria. Equivariant dimensions of groups with operators. Groups, geometry, and dynamics, Tome 16 (2022) no. 3, pp. 1049-1075. doi: 10.4171/ggd/686
@article{10_4171_ggd_686,
author = {Mark Grant and Ehud Meir and Irakli Patchkoria},
title = {Equivariant dimensions of groups with operators},
journal = {Groups, geometry, and dynamics},
pages = {1049--1075},
year = {2022},
volume = {16},
number = {3},
doi = {10.4171/ggd/686},
url = {http://geodesic.mathdoc.fr/articles/10.4171/ggd/686/}
}
TY - JOUR
AU - Mark Grant
AU - Ehud Meir
AU - Irakli Patchkoria
TI - Equivariant dimensions of groups with operators
JO - Groups, geometry, and dynamics
PY - 2022
SP - 1049
EP - 1075
VL - 16
IS - 3
UR - http://geodesic.mathdoc.fr/articles/10.4171/ggd/686/
DO - 10.4171/ggd/686
ID - 10_4171_ggd_686
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%0 Journal Article
%A Mark Grant
%A Ehud Meir
%A Irakli Patchkoria
%T Equivariant dimensions of groups with operators
%J Groups, geometry, and dynamics
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%P 1049-1075
%V 16
%N 3
%U http://geodesic.mathdoc.fr/articles/10.4171/ggd/686/
%R 10.4171/ggd/686
%F 10_4171_ggd_686
