Finite-volume hyperbolic 4-manifolds sharing a fundamental polyhedron
by
Dubravko Ivanšić
University of Oklahoma

It is known that the volume function for hyperbolic manifolds of dimension >= 3 is finite-to-one. We show that the number of nonhomeomorphic hyperbolic 4-manifolds with the same volume can be made arbitrarily large. This is done by constructing a sequence of finite-sided finite-volume polyhedra with side-pairings that yield manifolds. In fact, we show that arbitrarily many nonhomeomorphic hyperbolic 4-manifolds may share a fundamental polyhedron. As a by-product of our examples, we also show in a constructive way that the set of volumes of hyperbolic 4-manifolds contains the set of even integral multiples of 4\pi²/3. This is "half" the set of possible values for volumes, which is the integral multiples of 4\pi²/3 due to the Gauss-Bonnet formula \operatornameVol(M) = 4\pi²/3 ·\chi(M).

Date received: February 27, 1998

Copyright © 1998 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts. Document # cabb-02.