If M is a manifold with an action of a group G, then the homology group H_1(M,Q) is naturally a Q[G]-module, where Q[G] denotes the rational group ring. We prove that for every finite group G, and for every Q[G]-module V, there exists a closed hyperbolic 3-manifold M with a free G-action such that the Q[G]-module H_1(M,Q) is isomorphic to V. We give an application to spectral geometry: for every finite set P of prime numbers, there exist hyperbolic 3-manifolds N and N' that are strongly isospectral such that for all p in P, the p-power torsion subgroups of H_1(N,Z) and of H_1(N',Z) have different orders. We also show that, in a certain precise sense, the rational homology of oriented Riemannian 3-manifolds with a G-action "knows" nothing about the fixed point structure under G, in contrast to the 2-dimensional case. The main geometric techniques are Dehn surgery and, for the spectral application, the Cheeger-Muller formula, but we also make use of tools from different branches of algebra, most notably of regulator constants, a representation theoretic tool that was originally developed in the context of elliptic curves.

中文翻译：

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3-流形同源中的群表示

如果 M 是具有群 G 作用的流形，那么同调群 H_1(M,Q) 自然是一个 Q[G]-模，其中 Q[G] 表示有理群环。我们证明，对于每个有限群 G 和每个 Q[G]-模 V，存在一个闭双曲 3-流形 M，具有自由 G-作用，使得 Q[G]-模 H_1(M,Q)与 V 同构。 我们给出谱几何的应用：对于素数的每个有限集 P，存在强同谱的双曲 3-流形 N 和 N'，使得对于 P 中的所有 p，p 次幂扭转子群H_1(N,Z) 和 H_1(N',Z) 的顺序不同。我们还表明，在某种精确意义上，与二维情况相反，具有 G 作用的有向黎曼 3 流形的有理同调“不知道”G 下的不动点结构。