Simplicial complex with prescribed fundamental group

Question

Let $G$ be a finitely presented group. Can we find a finite simplicial complex $X$ such that $\pi_1(X)=G$ and $\pi_2(X)=0$?

I know the conclusion is true if we only require $X$ to be a CW-complex.

Answer 1

Yes, Hatcher constructs an infinite simplicial complex BG which is aspherical and has $\pi_1 BG = G$ in the appendices to Chapter 1 of his algebraic topology book.

Now given a simplicial complex $X$ with k-skeleton $X_k$, the map $i: X_k \to X$ induces an isomorphism on $\pi_i$ with $i < k$ and a surjection on $\pi_k$. This follows quickly from the cellular approximation theorem and its relative version, and holds for the k-skeleton of any CW complex; a simplicial complex is a special case.

So take the 3-skeleton $(BG)_3$. This has approximately $|G|^3$ simplices.