Counting hyperbolic manifolds
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.
scientific article; zbMATH DE number 1925566