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Volume of Seifert representations for graph manifolds and their finite covers

Pierre Derbez, Yi Liu, Shicheng Wang

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 $π^2$. 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.

Volume of Seifert representations for graph manifolds and their 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 . 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.

Paper Structure

This paper contains 37 sections, 37 theorems, 179 equations.

Table of Contents

  1. Introduction
  2. Summary of results
  3. Ingredients of proofs
  4. Organization
  5. The motion group of Seifert geometry
  6. Construction via central extensions
  7. Classification of conjugacy classes
  8. Construction of the Seifert geometry
  9. Volume of motion representations in Seifert geometry
  10. Volume of representations
  11. Seifert geometric circle bundles
  12. Topology of graph manifolds
  13. Seifert fiber spaces
  14. Graph manifolds
  15. Covering graph manifolds
  16. ...and 22 more sections

Key Result

Lemma 2.1

The expression defines a continuous map $\bar{\omega}\colon \mathrm{PSL}(2,{\mathbb R})\to {\mathbb R}/\pi{\mathbb Z}$, which is invariant under conjugation of $\mathrm{PSL}(2,{\mathbb R})$. Moreover, there exists a unique continuous, conjugation-invariant function $\widetilde{\omega}\colon {\widetilde{\mathrm{SL} for all $r\in{\mathbb R}$.

Theorems & Definitions (76)

  • Example 1.1: Twisted doubling
  • Example 1.2
  • Example 1.3
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Remark 2.5
  • ...and 66 more