Volumes of hyperbolic three-manifolds. (English) Zbl 0589.57015

Neumann, Walter D.; Zagier, Don

Topology 24, 307-332 (1985).

The authors introduce for a hyperbolic 3-manifold M with h cusps and for the manifold \(M_ K=M_{(p_ 1,q_ 1,...,p_ h,q_ h)}\) obtained by doing a \((p_ i,q_ i)\)-Dehn surgery on the i-th cusp some positive definite binary quadratic forms \(Q_ 1,...,Q_ n:\) \(Q_ i(p,q)=(length\) of \(pm_ i+q\ell_ i)^ 2/(vol T_ i)\). Here \(T_ i\) is a torus cross section of the i-th cusp. One of the authors’ main results: With the above notations, \[ vol M_ K=vol M-\pi^ 2\sum^{h}_{i=1}1/Q_ i(p_ i,q_ i)+O(\sum 1/(p^ 4_ i+q\quad^ 4_ i)). \] Note that the difference of volumes depends to a high order only on the geometry at the cusps and not on the rest of M. Corollary. Let \(\{M_{\nu}\}\) be the set of all hyperbolic 3-manifolds obtained from a given hyperbolic 3-manifold M by doing Dehn surgery on a single cusp of M. Then \[ \#\{\nu: Vol(M_{\nu})<Vol(M)-1/x\}=6\pi x+O(x^{1/2}) \] as \(x\to \infty\). If the Riemann hypothesis is true, then the exponent 1/2 can be replaced by \(67/148+\epsilon\) for any \(\epsilon >0\). Another main result of the paper gives some analytic aspects of a question related to the Chern-Simons invariant of M and to a holomorphic parametrization of deformations of the hyperbolic structures on M [no longer complete - see W. Thurston, ”The geometry and topology of 3-manifolds”, Mimeographed Lecture Notes, Princeton (1977/78); the reviewer, Z. Anal. Anwend. 5, 99-110 (1986; Zbl 0581.58040)].

Reviewer: B.N.Apanasov

MSC:

57N10	Topology of general \(3\)-manifolds (MSC2010)
58H15	Deformations of general structures on manifolds
53C30	Differential geometry of homogeneous manifolds
51M10	Hyperbolic and elliptic geometries (general) and generalizations

Keywords:

hyperbolic 3-manifold; cusps; Dehn surgery; positive definite binary quadratic forms; volumes; geometry at the cusps; Riemann hypothesis

Citations:

Zbl 0581.58040