On Algebraic and Geometric Dimensions for Groups With Torsion

Abstract

Various notions of dimension for discrete groups are compared. A group is exhibited that acts with finite stabilizers on an acyclic 2-complex in such a way that the fixed point subcomplex for any non-trivial finite subgroup is contractible, but such that the group does not admit any such action on a contractible 2-complex. This group affords a counterexample to a natural generalization of the Eilenberg-Ganea conjecture.

Equivariant CW-complexes and the orbit category

“…This theory was established by Bredon [6], tom Dieck [10] and Lück [20], and further developed by many authors (see, for example, Jackowski-McClure-Oliver [17, §5], Brady-Leary-Nucinkis [5], Symonds [34], [35], Grodal [14], Grodal-Smith [15]). In particular, the category of RΓ G -modules is an abelian category with Hom and tensor product, and has enough projectives for standard homological algebra.…”

“…For p = 5, the situation is easier than the case p = 3. Let R = Z (5) . The 5-period of S 5 equals 8, so by Swan [32] there exists a periodic projective resolution P over the group ring RG, giving an exact sequence 0 → R → P n → · · · → P 1 → P 0 → R → 0 for any positive integer n such that n + 1 = 3(k + 1) for some integer k, with 8 | (k + 1).…”

“…So, the above resolution is a projective resolution of I K R. We can also write a finite projective resolution for L (similar to the resolution given for N). So, by Proposition 6.8, we can replace C with a finite projective chain complex C (5) which has the desired homology.…”

On dimensions in Bredon homology

“…[22]. The definitions can easily be extended to infinite groups G and have recently regained attention through their use in equivariant obstruction theory [17] and through their connection to group actions on proper G-spaces [13,18,2,12,19]. Bredon homology also features in the Baum-Connes conjecture; a nice exposition of how the two relate can be found in [18].…”

“…If either dimension is one, it follows from a result of Dunwoody [7] that they are equal. In fact, it turns out [2] that there are examples of groups where cdG = 2 yet gdG = 3.…”