Taut foliations and the action of the fundamental group on leaf spaces and universal circles

Yosuke Kano

Vol. 267 (2014), No. 2, 399–416

DOI: 10.2140/pjm.2014.267.399

Abstract

Let $\mathcal{ℱ}$ be a leafwise hyperbolic taut foliation of a closed $3$-manifold $M$ and let $L$ be the leaf space of the pullback of $\mathcal{ℱ}$ to the universal cover of $M$. We show that if $\mathcal{ℱ}$ has branching, then the natural action of ${\pi }_1\left(M\right)$ on $L$ is faithful. We also show that if $\mathcal{ℱ}$ 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 ${\pi }_1\left(M\right)$.

Keywords

foliation, leaf space, universal circle

Mathematical Subject Classification 2010

Primary: 57M05, 57M60, 57R30

Milestones

Received: 15 March 2012

Revised: 16 July 2013

Accepted: 25 August 2013

Published: 11 May 2014