Critical Exponent and Displacement of Negatively Curved Free Groups

Abstract

We study the action of the fundamental group $\Gamma$ of a negatively curved 3-manifold $M$ on the universal cover $\tilde{M}$ of $M$. In particular we consider the ergodicity properties of the action and the distances by which points of $\tilde{M}$ are displaced by elements of $\Gamma$. 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 Poincaré series for $\Gamma$. For the case in which $n= 3$, assuming that $\Gamma$ is free of rank $k \geq 2$, that the limit set of $\Gamma$ has positive 2-dimensional Hausdorff measure, that $D = 2$ and that the Poincaré series diverges at the exponent 2, we prove a displacement estimate for $\Gamma$ which is identical to the one given by the log$(2k-1)$ theorem [1] for the constant-curvature case.