## Journal of Differential Geometry

- J. Differential Geom.
- Volume 57, Number 1 (2001), 173-193.

### 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.

#### Article information

**Source**

J. Differential Geom., Volume 57, Number 1 (2001), 173-193.

**Dates**

First available in Project Euclid: 20 July 2004

**Permanent link to this document**

https://projecteuclid.org/euclid.jdg/1090348091

**Digital Object Identifier**

doi:10.4310/jdg/1090348091

**Mathematical Reviews number (MathSciNet)**

MR1871493

**Zentralblatt MATH identifier**

1038.53040

#### Citation

Hou, Yong. Critical Exponent and Displacement of Negatively Curved Free Groups. J. Differential Geom. 57 (2001), no. 1, 173--193. doi:10.4310/jdg/1090348091. https://projecteuclid.org/euclid.jdg/1090348091