Counting Hyperbolic Manifolds with Bounded Diameter |
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Authors: | Robert Young |
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Affiliation: | (1) Department of Mathematics, University of Chicago, 5734S University Avenue, Chicago, IL, 60637, U.S.A |
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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. |
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Keywords: | diameter hyperbolic manifolds |
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