Counting hyperbolic manifolds with bounded diameter

Research output: Contribution to journalArticle

Abstract

Let ρ n (V) be the number of complete hyperbolic manifolds of dimension n with volume less than V. Burger et al [Geom. Funct. Anal. 12(6) (2002), 1161-1173.] showed that when n ≥ 4 there exist a, b > 0 depending on the dimension such that aV log V ≤ log ρ n (V) ≤ bV log V, for V ≫ 0. In this note, we use their methods to bound the number of hyperbolic manifolds with diameter less than d and show that the number grows double-exponentially with volume. Additionally, this bound holds in dimension 3.

Original languageEnglish (US)
Pages (from-to)61-65
Number of pages5
JournalGeometriae Dedicata
Volume116
Issue number1
DOIs
StatePublished - Dec 2005

Keywords

  • Diameter
  • Hyperbolic manifolds

ASJC Scopus subject areas

  • Geometry and Topology

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