Counting hyperbolic manifolds with bounded diameter

Details Counting hyperbolic manifolds with bounded diameter Counting hyperbolic manifolds with bounded diameter

2006-01-23

arxiv.org/abs/math/0601560v1

Geometriae Dedicata, 116(2005), 61 - 65

Mathematics

Differential Geometry

4 pages

Scientific paper

Let $\rho_n(V)$ be the number of complete hyperbolic manifolds of dimension n with volume less than $V$. Burger, Gelander, Lubotzky, and Moses showed that when n>3 there exist a,b>0 depending on the dimension such that aV log(V) < log(\rho_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. Additionally, this bound holds in dimension 3.

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