For any closed orientable 3-manifold, there is a volume function defined on the space of all Seifert representations of the fundamental group. The maximum absolute value of this function agrees with the Seifert volume of the manifold due to Brooks and Goldman. For any Seifert representation of a graph manifold, the authors establish an effective formula for computing its volume, and obtain restrictions to the representation as analogous to the Milnor–Wood inequality (about transversely projective foliations on Seifert fiber spaces). It is shown that the Seifert volume of any graph manifold is a rational multiple of . Among all finite covers of a given nongeometric graph manifold, the supremum ratio of the Seifert volume over the covering degree can be a positive number, and can be infinite. Examples of both possibilities are discovered, and confirmed with the explicit values determined for the finite ones.

中文翻译：

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图流形及其有限覆盖的 Seifert 表示体积

对于任何闭可定向 3 流形，在基本群的所有 Seifert 表示的空间上定义有一个体积函数。该函数的最大绝对值与 Brooks 和 Goldman 提出的流形 Seifert 体积一致。对于图流形的任何 Seifert 表示，作者建立了一个有效的公式来计算其体积，并获得了类似于 Milnor-Wood 不等式（关于 Seifert 纤维空间上的横向射影叶状结构）的表示限制。结果表明，任何图流形的 Seifert 体积都是有理数倍。在给定的非几何图流形的所有有限覆盖中，Seifert体积与覆盖度的上确比可以是正数，并且可以是无穷大。这两种可能性的例子都被发现了，并用为有限值确定的显式值进行了确认。