@YCor: great idea to get it down to $F_4$.

Sorry I didn't see your question at the time. If you have a finite $n$-dimensional simplicial complex and the link of every vertex is an $(n-1)$-dimensional homology manifold then each $(n-1)$-simplex is a face of exactly two $n$-simplices, so the mod-2 chain consisting of the sum of all the $n$-simplices is a cycle. Even without knowing that the vertex links are homology spheres this shows that $H_n(K:\mathbb{F}_2)\neq 0$.

@SebastienPalcoux: My point was that it is very hard to say what one means by good. I just had a quick look at your idea of irreducible $n$-blocks. In the case $n=1$ it has something in common with Brian Bowditch's idea of a taut loop in the Cayley graph (see his article `Continuously many quasi-isometry classes of 2-generator groups'). For larger $n$ I think there is a problem: why should (for example) a 2-block be a 2-sphere rather than say a 2-torus or some other surface?

@SebastienPalcoux: Knowing a good set of generators for a group doesn't help you to find a good set of relators, so why should fixing a group presentation help you to find a good set of attaching maps for 3-cells?

Isn't what you are looking for just the $n$-skeleton of a model for $EG$?

Isn't the 1-dimensional case the source of all of this? The general linear group $GL_1(\mathbb{R})$ consists of all non-zero real numbers and has two components. If you believe in the existence and some properties of the determinant then the same follows for $GL_n(\mathbb{R})$ for all $n$.

The construction $Y\mapsto T(Y)$ can be designed in such a way that it is natural for maps of simplicial complexes that do not collapse any simplices; in particular for simplicial automorphisms. The free action of $\pi_1(X)$ on $\widetilde{X}$ thus induces an action of $\pi_1(X)$ on $T(\widetilde{X})$ too.

This uses the Morse function on the universal covering space of the Salvetti complex, which gets called $X_L$. The RAAG $A_L$ acts freely cocompactly on $X_L$, and $BB_L$ acts freely cocompactly on the level set (= inverse image of a point under the Morse function $f:X_L\rightarrow \mathbb{R}$). The whole of $X_L$ is built by attaching cones on $L$ to the level set. In the case when $L$ is acyclic, it follows that the level set is itself acyclic, since it must have the same homology as $X_L$ which is contractible. Thus the dimension of the level set is an upper bound for $cd(BB_L)$.