Finitely dominated universal spaces for the family of solvable subgroups

Question

$\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\Sz{Sz}$In short, I am interested in the question which finite groups $G$ admit a finitely dominated universal space with respect to the family of solvable subgroups. That is, for which finite groups does there exist a finitely dominated $G$-CW complex $X$ such that for a subgroup $H \leq G$ one has

• $X^H$ is contractible if $H$ is solvable;
• $X^H$ is empty if $H$ is not solvable.

Of course, if $G$ is solvable then the point has all the desired properties. More examples can be found in the literature. I will first mention Adem's general work and then a classical construction of Floyd–Richardson.

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In his article Finite group actions on acyclic $2$-complexes, Adem answers the question which finite simple groups $G$ can act on a finite acyclic $2$-complex without fixed points. He constructs explicit examples of $G$-complexes $X$ such that $X^H$ is acyclic if $H$ is solvable, and empty if $H$ is not solvable. Taking the join of such an example with itself (that is, the homotopy pushout of $X \leftarrow X \times X \rightarrow X$) gives a finite model for the universal space for the family of solvable subgroups of $G$.

In total, Adem's result show that a finite model exists for three infinite families of finite groups. They are:

• $\PSL_2(2^k)$ for $k$ at least $2$;
• $\PSL_2(q)$ for $q \geq 5$ congruent to $\pm3$ mod $8$;
• $\Sz(2^k)$ for $k\geq 3$ odd.

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Using work of Floyd–Richardson, one can construct a finite $A_5$-CW complex modeling the universal space for the family of solvable subgroups. Note that every proper subgroup of $A_5$ is solvable. The alternating group $A_5$ includes into $\operatorname{SO}(3)$ as the icosahedral subgroup. The coset space $\operatorname{SO}(3)/A_5$ (the Poincaré homology sphere) carries a left $A_5$-action with a single fixed point. Taking out that fixed point and collapsing to a $2$-skeleton gives rise to an $A_5$-action on an acyclic $2$-complex without fixed point. Smith-theoretic methods show that each solvable subgroup of $A_5$ has to have a fixed point, and the fixed point space is acyclic as well (see Thm. 3.1. in Adem's aforementioned article). Again, take the join of this $A_5$-space with itself.

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In total, it seems like for quite a few finite simple groups $G$ there is a finitely dominated $G$-space modeling the universal space for the family of solvable subgroups. Are there more examples? Is there a finite group $G$ for which there exists no such space?

Answer 1

The following was brought to my attention by Christoph Winges. In his work in smooth group actions on discs, Bob Oliver shows that for a compact Lie group $G$ there exists a smooth action on a disc such that for each $H\leq G$ the fixed set is

• a disc if $H$ is an extension of a torus by a solvable group;
• empty otherwise.

He mentions in the introduction that by other work of his, constructing such actions on discs is equivalent to constructing some finite $G$-CW complex of the same equivariant homotopy type.

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The precise reference is Bob Oliver's article Smooth compact Lie group actions on discs, Theorem 4 for a more general statement about certain families which are closed and separating combined with Proposition 8 showing that the aforementioned family is closed and separating.

Answer 2

This happens if and only if the Weyl group of every solvable subgroup is trivial.

For one direction, use the tom Dieck splitting and the fact that the classifying space of a nontrivial finite group is never finitely dominated.

Conversely, if these Weyl groups are all trivial then $G/H$ has no automorphisms as a $G$-space. Inducting up the lattice if subgroups, we see that your space is a finite colimit of representanbles as a pre sheaf on the orbit category. So it’s not only finitely dominated— it’s finite.