Critical exponent and displacement of negatively curved free groups.
type: Item
instance of: scholarly article
title: Critical exponent and displacement of negatively curved free groups.
zbMATH Open document ID: 1038.53040
author: Yong Hou
published in: Journal of Differential Geometry
publication date: 2002-08-15T00:00:00Z
review text: 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…
reviewed by: Darryl McCullough
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 51M10
Mathematics Subject Classification ID: 30F40
Mathematics Subject Classification ID: 22F50
Mathematics Subject Classification ID: 53A35
zbMATH DE Number: 1782676
zbMATH Keywords: pinched negative curvature
zbMATH Keywords: group
zbMATH Keywords: free
zbMATH Keywords: Busemann metric
zbMATH Keywords: Gromov metric
zbMATH Keywords: shadow metric
zbMATH Keywords: critical exponent
zbMATH Keywords: Patterson-Sullivan theory
zbMATH Keywords: conformal density
zbMATH Keywords: Poincaré series
zbMATH Keywords: displacement
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: Topogonov comparison theorem
zbMATH Keywords: harmonic measure
zbMATH Keywords: Culler-Shalen paradoxical decomposition theorem
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1090348091
OpenAlex ID: W2137579813
Wikidata QID: Q115202872
DOI: 10.4310/JDG/1090348091
description: scientific article; zbMATH DE number 1782676