Taut foliations and the actions of fundamental groups on leaf spaces and universal circles

TL;DR

This paper investigates the actions of fundamental groups on leaf spaces and universal circles in the context of taut foliations, revealing conditions under which these actions are faithful and characterizing stabilizers of branch loci.

Contribution

It establishes that for leafwise hyperbolic taut foliations with branching, the fundamental group acts faithfully on the leaf space, and describes the structure of stabilizers of finite branch loci.

Findings

Fundamental group acts faithfully on leaf space if foliation has branching.

Stabilizer of a finite branch locus is an infinite cyclic group.

Generated by an indivisible element of the fundamental group.

Abstract

Let $F$ be a leafwise hyperbolic taut foliation of a closed 3-manifold $M$ and let $L$ be the leaf space of the pullback of $F$ to the universal cover of $M$. We show that if $F$ has branching, then the natural action of $\pi_1(M)$ on $L$ is faithful. We also show that if $F$ has a finite branch locus $B$ whose stabilizer acts on $B$ nontrivially, then the stabilizer is an infinite cyclic group generated by an indivisible element of $\pi_1(M)$.