Counting hyperbolic manifolds
type: Item
instance of: scholarly article
title: Counting hyperbolic manifolds
zbMATH Open document ID: 1029.57021
published in: Geometric and Functional Analysis. GAFA
publication date: 2003-07-02T00:00:00Z
full work available at URL: http://hdl.handle.net/20.500.11850/422691
review text: A classical theorem of Wang implies that for a fixed dimension \(n \geq 4\) and any \(V \in \mathbb R\), there are only finitely many complete hyperbolic manifolds without boundary of volume at most \(V\) up to isometries. In this interesting paper, the authors obtain upper and lower bounds for the number \(\rho_n(V)\) of such manifolds with volume at most \(V\). Specifically, they prove: \textbf{Theorem}. For every \(n \geq 4\), there are two constants \(a=a(n)>0\) and \(b=b(n)>0\) such that for all sufficiently large \(V\), \[ aV\log V \leq \log \rho_n(V) \leq bV\log V. \] In particular, this implies that for \(n \geq 4\) the growth rate of \(\rho_n(V)\) is given by \(\log \rho_n(V) \approx V \log V\). The proof of the lower bound uses results of Gromov and Piatetski-Shapiro, as well as Lubotzky and Margulis. The bulk of the paper is devoted to the proof of the upper bound which is somewhat technical and uses the thick-thin decomposition of a complete hyperbolic manifold and the construction of a vector field transversal to the boundary of the thin part. The paper also contains some remarks about the problem of estimating the growth rate of the number of conjugacy classes of irrreducible lattices of covolume at most \(V\) in a semisimple Lie group \(G\) without compact factors (\(G\) not locally isomorphic to \(SL_2(\mathbb R)\) or \(SL_2(\mathbb C)\)), which generalises the above problem.
Mathematics Subject Classification ID: 57N16
Mathematics Subject Classification ID: 22E40
zbMATH DE Number: 1925566
author: Marc Burger
author: Tsachik Gelander
author: Alexander Lubotzky
author: Shahar Mozes
reviewed by: Ser Peow Tan
MaRDI profile type: Publication
OpenAlex ID: W2119776512
DOI: 10.1007/S00039-002-1161-1
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description: scientific article; zbMATH DE number 1925566