Critical exponent and displacement of negatively curved free groups. (Q1609811)

Let \(M\) be a complete Riemannian manifold whose curvature satisfies \(-1\leq \kappa \leq -a\) for some \(a>0\). The fundamental group \(\Gamma\) of \(M\) acts as isometries on the universal cover \(\widetilde{M}\), and this action extends to the sphere at infinity \(S_\infty\). Assume that \(\Gamma\) is finitely generated and free, and let \({\mathcal S}\) be an arbitrary generating set of \(\Gamma\), where \({\mathcal S}\) does not include inverses. Define \(D\) to be the critical exponent of the Poincaré series \(\sum_{\gamma\in\Gamma} \exp(-s\, \text{dist}(x, \gamma(x)))\), and let \[ \text{Disp}(r):=\sum_{\gamma\in {\mathcal S}}{{1}\over{{1+\exp(r\, \text{dist}(x, \gamma(x)))}}}. \] The main results of this paper are that for any \(x\): (1) \(\text{Disp}(D)\leq 1/2\), and (2) if \(M\) has dimension \(3\), \(D=2\), \(\Gamma\) is divergent (that is, the Poincaré series diverges at \(s=D\)), and the \(2\)-dimensional Hausdorff measure on \(S_\infty\) with respect to the Busemann metric is positive, then \(\text{Disp}(1)\leq 1/2\). The expression \(\text{Disp}(1)\) is called the displacement, and the inequality in (2) easily implies that at least one of the \(\text{dist}(x, \gamma(x))\) is greater than or equal to \(\ln(2k-1)\), where \(k\) is the rank of \(\Gamma\). In the case of constant negative curvature, this \(\ln(2k-1)\) bound was obtained by \textit{M. Culler} and \textit{P. B. Shalen} [J. Am. Math. Soc. 5, 231--288 (1992; Zbl 0769.57010)] for \(k=2\), and by \textit{J. Anderson}, \textit{R. Canary}, \textit{M. Culler}, and \textit{P. B. Shalen} [J. Differ. Geom. 43, 738--782 (1996; Zbl 0860.57011)] for \(k\geq 2\). When \(D<1\), result (1) gives a better bound than \(\ln(2k-1)\). Another application of result (1) is that if \(M\) is also a rank-\(1\) locally symmetric space, then \(\text{Disp}(d)\leq 1/2\) where \(d\) is the Hausdorff dimension of the limit set of \(\Gamma\). Basic tools for the proofs of the main results include the Toponogov Comparison Theorem and various metrics on \(S_\infty\), such as the Busemann metric, the Gromov metric, and the shadow metric. Among the technical results developed in the paper are a generalization of the Culler-Shalen paradoxical decomposition theorem to the case of pinched negative curvature, and a development of the Patterson-Sullivan theory in this setting.