The Thurston norm of graph manifolds

TL;DR

This paper characterizes the Thurston norm of graph manifolds as sums of absolute linear functionals, showing all such norms with rational coefficients are realizable, and explores the geometric structure of their unit balls.

Contribution

It proves that any rational-sum of absolute value linear functional norm can be realized by a graph manifold, and examines the geometric properties of these norms' unit balls.

Findings

Every rational-sum of absolute value linear functionals is realizable as a Thurston norm of some graph manifold.

All symmetric polygons with rational vertices can be the unit ball of a graph manifold fibered over the circle.

Certain symmetric polyhedra are not realizable as Thurston norm balls, but can be refined by graph manifolds.

Abstract

The Thurston norm of a closed oriented graph manifold is a sum of absolute values of linear functionals, and either each or none of the top-dimensional faces of its unit ball are fibered. We show that, conversely, every norm that can be written as a sum of absolute values of linear functionals with rational coefficients is the nonvanishing Thurston norm of some graph manifold, with respect to a rational basis on its second real homology. Moreover, we can choose such graph manifold either to fiber over the circle or not. In particular, every symmetric polygon with rational vertices is the unit polygon of the nonvanishing Thurston norm of a graph manifold fibering over the circle. In dimension $\ge 3$ many symmetric polyhedra with rational vertices are not realizable as nonvanishing Thurston norm ball of any graph manifold. However, given such a polyhedron, we show that there is always a graph manifold whose nonvanishing Thurston norm ball induces a finer partition into cones over the faces.