We calculate the Bieri–Neumann–Strebel–Renz invariant Σ1(G) for finitely presented residually free groups G and show that its complement in the character sphere S(G) is a finite union of finite intersections of closed sub-spheres in S(G). Furthermore, we find some restrictions on the higher-dimensional homological invariants Σn(G, ℤ) and show for the discrete points Σ2(G)dis, Σ2(G, ℤ)dis and Σ2(G, ℚ)dis in Σ2(G), Σ2(G, ℤ) and Σ2(G, ℚ) that we have the equality Σ2(G)dis = Σ2(G, ℤ)dis = Σ2(G, ℚ)dis.

中文翻译：

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关于剩余自由群的 Bieri-Neumann-Strebel-Renz 不变量

我们计算 Bieri–Neumann–Strebel–Renz 不变量 Σ1(G) 对于有限呈现的剩余自由群G并表明它在字符范围内的补码小号(G) 是封闭子球体的有限交集的有限并集小号(G）。此外，我们发现对高维同调不变量 Σ 的一些限制n(G, ℤ) 并显示离散点 Σ2(G)迪斯, Σ2(G, ℤ)迪斯和 Σ2(G, ℚ)迪斯在 Σ2(G), Σ2(G, ℤ) 和 Σ2(G, ℚ) 我们有等式 Σ2(G)迪斯= Σ2(G, ℤ)迪斯= Σ2(G, ℚ)迪斯.