Fix a finite group $G$. We study the computational complexity of counting problems of the following flavor: given a group $Γ$, count the number of homomorphisms $Γ\to G$. Our first result establishes that this problem is $\#\mathsf{P}$-hard whenever $G$ is a non-abelian group and $Γ$ is provided via a finite presentation. We give several improvements showing that this hardness conclusion continues to hold for restricted $Γ$ satisfying various promises. Our second result shows that if $G$ is class 2 nilpotent and $Γ= π_1(M^3)$ 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 $Γ$ 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 $Γ$ is itself a finite group encoded via a multiplication table, provided that $|H^2(Γ,Z(G))|$ is similarly bounded from above.

翻译：固定一个有限群$G$。我们研究下列计数问题的计算复杂性：给定一个群$Γ$，计算同态$Γ\to G$的数量。我们的第一个结果表明，只要$G$是非阿贝尔群且$Γ$通过有限表示给出，该问题即为$\#\mathsf{P}$-难的。我们给出若干改进，表明这一困难性结论对于满足特定约定的受限制$Γ$仍然成立。第二个结果表明，若$G$是2类幂零群且$Γ=π_1(M^3)$（其中$M^3$为输入的三维流形三角剖分，且$|H^2(M,Z(G))|$有上界），则存在多项式时间算法计算从$Γ$到$G$的同态数量。该算法的部分合理性源于三维流形足够接近Eilenberg-MacLane空间，使得我们能利用给定的三角剖分高效解决必要的群上同调障碍问题。当$Γ$本身是通过乘法表编码的有限群，且$|H^2(Γ,Z(G))|$同样具有上界时，针对有限2类幂零群$G$的同态计数也存在类似的多项式时间算法。