Let $X= X(P)$ be the Cayley complex of a finite group presentation $P=<S | R>$. Are there geometric properties of $X$ that are known to be decidable by an algorithm that takes $P$ as input? For example, being a tree is trivially decidable. I think I can prove that being planar is also decidable.
The question makes more sense for properties that depend on the choice of $P$ and not only on the group, so that e.g. the Adjan-Rabin theorem cannot be applied.