TITLE: Small simplicial set models for BG
QUESTION [12 upvotes]: 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.

REPLY [4 votes]: 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.