Walter D. Neumann

Volumes of hyperbolic three-manifolds

BY“hyperbolic 3-manifold” we will mean an orientable complete hyperbolic 3-manifold M of finite volume. By Mostow rigidity the volume of M is a topological invariant, indeed a homotopy invariant, of the manifold M. There is in fact a purely topological definition of this invariant, due to Gromov. The set of all possible volumes of hyperbolic 3-manifolds is known to be a well-ordered subset of the real numbers and is of considerable interest (for number theoretic aspects see, for instance, [2], [13]) but remarkably little is known about it: the smallest element is not known even approximately, and it is not known whether any element of this set is rational or whether any element is irrational. For more details see Thurston’s Notes [73. In this paper we prove a result which, among other things, gives some metric or analytic information about the set of hyperbolic volumes. Given a hyperbolic 3-manifold M with h cusps, one can form the manifold MK = M(P1.41. . , Ph4J obtained by doing a (pi, q,)-Dehn surgery on the i-th cusp, where (pi, qi) is a coprime pair of integers, or the symbol 03 if the cusp is left unsurgered. This notation is well defined only after choosing a basis mi, di for the homology HI (Q, where Z is a torus cross section of the i-th cusp. Then (pi, q,)-Dehn surgery means: cut off the i-th cusp and paste in a solid torus to kill Pimi+qiei.

Extended Bloch group and the Cheeger-Chern-Simons class

We dene an extended Bloch group and show it is naturally isomorphic to H3(PSL(2;C);Z). Using the Rogers dilogarithm function this leads to an exact simplicial formula for the universal Cheeger{Chern{Simons class on this homology group. It also leads to an independent proof of the analytic relationship between volume and Chern{Simons invariant of hyperbolic 3{manifolds conjectured in [16] and proved in [24], as well as eective formulae for the Chern{Simons invariant of a hyperbolic 3{manifold. AMS Classication numbers Primary: 57M27 Secondary: 19E99, 57T99

Computing arithmetic invariants of 3-manifolds

Snap is a computer program for computing arithmetic invariants of hyperbolic 3-manifolds, built on Jeff Weekss SnapPea and the number theory package Pari. Its approach is to compute the hyperbolic structure to very high prec ision, and use th is to find an exact description of the structure. Then the correctness of the hyperbolic structure can be verified, and the arithmetic invariants of Neumann and Reid can be computed. Snap also computes high precision numerical invariants such as volume, Chern–Simons invariant, eta invariant, and the Borel regulator.

Bloch invariants of hyperbolic

We define an invariant \beta(M) of a finite volume hyperbolic 3-manifold M in the Bloch group B(C) and show it is determined by the simplex parameters of any degree one ideal triangulation of M. \beta(M) lies in a subgroup of \B(\C) of finite \Q-rank determined by the invariant trace field of M. Moreover, the Chern-Simons invariant of M is determined modulo rationals by \beta(M). This leads to a simplicial formula and rationality results for the Chern Simons invariant which appear elsewhere. Generalizations of \beta(M) are also described, as well as several interesting examples. An appendix describes a scissors congruence interpretation of B(C).

3

SummaryWe show that the set

Automatic structures, rational growth, and geometrically finite hyperbolic groups

S\mathfrak{A}(G)

Canonical Decompositions of 3{Manifolds

of equivalence classes of synchronously automatic structures on a geometrically finite hyperbolic groupG is dense in the product of the sets