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Let $G$ be a discrete group that acts on a contractible finite dimensional $G$-complex $X$ with the following properties:

  • $X/G$ is compact (i.e. the action is cocompact)
  • Each stabilizer $G_\sigma$ admits a cocompact action on a contractible finite dimensional $G_\sigma$-complex with finite stabilizers

Question: Is there a finite dimensional contractible $G$-complex $Y$ with finite stabilizers such that $Y/G$ is compact ?

The question can be stated more conceptually by help of the following definition: Let $\mathscr{F}$ be the class of all finite groups and define the class $K_i\mathscr{F},\;i\ge 0$ inductively by

  • $K_0\mathscr{F} := \mathscr{F}$
  • $K_i\mathscr{F}$ includes all groups $G$ that admit a finite dimensional contractible $G$-complex $X$ such that (1) $X/G$ is compact and (2) the stabilizers are in $K_{i-1}\mathscr{F}$.

Then the question is equivalent to

Question: Is $K_1\mathscr{F} \subsetneqq K_2\mathscr{F}$ ?

Remark: If condition (1) is dropped, we get the classical Kropholler classes $H_i\mathscr{F}$. There (among many other results) $H_1\mathscr{F} \subsetneqq H_2\mathscr{F}$ is known: For example, the free abelian group of countably infinite rank belongs to $H_2\mathscr{F}\setminus H_1\mathscr{F}$.

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1 Answer 1

It follows from the proof of prop. 5.1 in

'Lück, W. and Weiermann, M.,On the classifying space of the family of virtually cyclic subgroups,Pure and Applied Mathematics Quarterly, Vol. 8(2) (2012), 497-555 (it's on the arxiv)'

that if $X$ is a cocompact $G$-CW-complex such that all its stabilizers have cocompact classifying space for proper actions, then $X \times E_{\mathcal{F}}G$ is $G$-homotopy equivalent to a cocompact $G$-CW-complex. Here, $E_{\mathcal{F}}G$ is any model for the classifying space for proper actions of $G$.

Hence, the answer to your question is yes, if you assume additionally that all stabilizers have cocompact classifying space for proper actions.

Without this assumption, I think this is an open problem.

On the other hand, every group in $K_2\mathcal{F}$ is of type $FP_{\infty}$. Therefore, it has a finite dimensional model for proper actions by a result of Kropholler and Mislin. Hence, $K_2\mathcal{F} \subseteq H_1\mathcal{F}$.

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Dieter, thanks, but I won't assume further restrictions on the fixed point spaces. Actually, I wonder, if the strict inclusion $H_1\mathscr{F} \subsetneqq H_2\mathscr{F}$ of the Kropholler classes also holds for cocompact actions. –  Ralph May 27 '13 at 20:40
    
Ralph, I've edited my answer to get rid of the assumptions on the fixed point spaces –  Dieter May 28 '13 at 8:08

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