Small simplicial set models for BG

Question

Let $F$ be a finite group.

Is there a model for $BF$ as a simplicial set such that the number of nondegenerate $n$-simplices grows at most polynomially?

For example the Bar construction has the property that there are exactly $(|F|-1)^n$ nondegenerate $n$-simplices. This answers the question affirmatively for $\mathbb{Z}/2$, but for other groups it still grows exponentially.

A lower bound for the number of such simplices is of course given by the rank of the group homology and in all examples that I know this only grows polynomially.

Of course it would be nice to have a functorial model, but that might be a follow up.

Answer 1

Going out on a limb (I may well be messing up badly), I think the answer is yes at least for the CW structure version of the question.

Proof:

Choose a finite presentation of $F$ with generating set $G$ and relation set $R$, and consider the induced $\pi_1$-isomorphism $X = (S^1)^{\vee G} \cup_{(S^1)^{\vee R}} \ast \to BF$. Note that $X$ has finitely many cells. I believe that the rank over $\mathbb Z[F]$ of $H_\ast(\tilde X ; \underline{\mathbb Z[F]})$ grows polynomially in $\ast$ (if it doesn't, then we should have a homology obstruction, giving a negative answer to the question). Then we should be able to mimic the construction showing that the homology bound on number of cells is realized for a simply-connected space. That is, we attach free $F$-equivariant cells (i.e. cells of the form $\vee^{F} D^n_+$, with $n \geq 2$) to $\tilde X$ one-by-one, building up a space $\tilde X'$ which has the same $\underline{\mathbb Z[F]}$-homology as $\tilde{BF}$ and conclude by homology Whitehead (with coefficients) that $X' \to BF$ is a weak homotopy equivalence, where $X' = \tilde X'_{hF}$ has one cell for each cell of $X$ plus a cell for each $F$-equivariant cell we attached.

(I'm not 100% sure though -- in the simply-connected case, we use the Hurewicz theorem to be sure that we can always map to a homology class with a sphere... perhaps this breaks down if the relevant $F$-equivariant Hurewicz theorem fails?)

Answer 2

There is an algebraic result that is relevant, due to Benson and Carlson and stated as Corollary 4.5 in `Complexity and Multiple Complexes' Math. Z. vol 195 (1987) 221--238. Given a finite group $G$, let $n$ be the maximum of the $p$-ranks of $G$ over all primes. Then there is a free resolution of $\mathbb{Z}$ over $\mathbb{Z}G$ that is the tensor product of $n$ non-negative periodic complexes. The sizes of the modules in this resolution grow as a polynomial of degree $n-1$, and one can think of the resolution as being built from copies of a finite chain complex of free $\mathbb{Z}G$-modules: the tensor product of the period pieces for the $n$ periodic complexes. This is the same sort of thing as you would get if the group acted freely trivially on homology on a product of $n$ spheres (possibly of different dimensions).

There are two issues with promoting this to the result that you want: firstly realizing the periodic pieces as the chain complexes of simply connected $G$-CW-complexes, for which Tim Campion's answer is relevant; secondly somehow realizing the whole complex as a simplicial set in such a way that you don't need any extra low-dimensional non-degenerate simplices in the higher-dimensional copies of the periodic pieces. For the second of these, the cyclic group $C_n$ (where the periodic piece should be the chain complex for the circle with the group acting freely by rotation) is an important test case.

Answer 3

This question may be somewhat relevant: Small simplicial complexes with torsion in their homology. David Speyer's answer there shows that one can build a simplicial complex $X$ with $H_1(X)=\mathbb{Z}/p$ where the number of simplices of $X$ is $O(\log(p))$. It seems unlikely that one can do much better than that in the world of simplicial sets. With CW complexes, you only need a single $0$-cell, a single $1$-cell and a single $2$-cell. This gives an initial picture of how the simplicial set version of the question might deviate from the CW complex version.