We study the action of the fundamental group $\G$ of a negatively curved $3$-manifold $M$ on the universal cover $\widetilde M$ of $M$. In particular we consider the ergodicity properties of the action and the distances by which points of $\widetilde M$ are displaced by elements of $\G$. First we prove a displacement estimate for a general $n$-dimensional manifold with negatively pinched curvature and free fundamental group. This estimate is given in terms of the critical exponent $D$ of the Poincar\'e series for $\G$. For the case in which $n=3$, assuming that $\G$ is free of rank $k\ge2$, that the limit set of $\G$ has positive $2$-dimensional Hausdorff measure, that $D=2$ and that the Poincar\'e series diverges at the exponent $2$, we prove a displacement estimate for $\G$ which is identical to the one given by the $\log(2k-1)$ theorem [ACCS] for the constant-curvature case.