Obstruction theory and the complexity of counting group homomorphisms

Fix a finite group $G$. We study the computational complexity of counting problems of the following flavor: given a group $\Gamma$, count the number of homomorphisms $\Gamma \to G$. Our first result establishes that this problem is $\#\mathsf{P}$-hard whenever $G$ is a non-abelian group and $\Gamma$ is provided via a finite presentation. We give several improvements showing that this hardness conclusion continues to hold for restricted $\Gamma$ satisfying various promises. Our second result shows that if $G$ is class 2 nilpotent and $\Gamma = \pi_1(M^3)$ for some input 3-manifold triangulation $M^3$ with $|H^2(M,Z(G)|$ bounded above, then there is a polynomial time algorithm to compute the number of homomorphisms from $\Gamma$ to $G$. This algorithm is explained in part by the fact that 3-manifolds are close enough to being Eilenberg-MacLane spaces for us to solve the necessary group cohomological obstruction problems efficiently using the given triangulation. A similar polynomial time algorithm for counting maps to finite, class 2 nilpotent $G$ exists when $\Gamma$ is itself a finite group encoded via a multiplication table, provided that $|H^2(\Gamma,Z(G))|$ is similarly bounded from above.