  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 $\pi_1(M)$.
