Large and small covers of a hyperbolic manifold

Petra Bonfert-Taylor, Katsuhiko Matsuzaki, Edward C. Taylor

Research output: Contribution to journalArticle

5 Citations (Scopus)

Abstract

The exponent of convergence of a non-elementary discrete group of hyperbolic isometries measures the Hausdorff dimension of the conical limit set. In passing to a non-trivial regular cover the resulting limit sets are point-wise equal though the exponent of convergence of the cover uniformization may be strictly less than the exponent of convergence of the base. We show in this paper that, for closed hyperbolic surfaces, the previously established lower bound of one half on the exponent of convergence of "small" regular covers is sharp but is not attained. We also consider "large" (non-regular) covers. Here large and small are descriptive of the size of the exponent of convergence.We show that a Kleinian group that uniformizes a manifold homeomorphic to a surface fibering over a circle contains a Schottky subgroup whose exponent of convergence is arbitrarily close to two.

Original languageEnglish
Pages (from-to)455-470
Number of pages16
JournalJournal of Geometric Analysis
Volume22
Issue number2
DOIs
Publication statusPublished - 2012 Apr 1
Externally publishedYes

Keywords

  • Bottom of spectrum
  • Conical limit set
  • Exponent of convergence
  • Geodesic flow
  • Kleinian groups

ASJC Scopus subject areas

  • Geometry and Topology

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