Volume of Seifert representations for graph manifolds and their finite covers

TL;DR

This paper provides a new effective formula for computing the Seifert volume of graph manifolds, explores restrictions similar to the Milnor--Wood inequality, and investigates the behavior of Seifert volume under finite covers, including cases where the ratio can be infinite.

Contribution

It introduces an explicit formula for Seifert volume of graph manifolds and analyzes its behavior under finite covers, revealing cases with unbounded ratios.

Findings

Seifert volume of graph manifolds is a rational multiple of π^2.

The supremum ratio of Seifert volume over degree can be positive or infinite.

Explicit values of Seifert volume are determined for certain finite covers.

Abstract

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 $\pi^2$. Among all finite covers of a given non-geometric 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.