Hyperbolic volume of representations of fundamental groups of cusped 3-manifolds

Details Hyperbolic volume of representations of fundamental groups of cusped 3-manifolds Hyperbolic volume of representations of fundamental groups of cusped 3-manifolds

2003-05-19

arxiv.org/abs/math/0305275v1

A revised version is published: Int. Math. Res. Not., (9):425--459, 2004

Mathematics

Geometric Topology

38 pages

Scientific paper

Let W be a compact manifold and let \rho be a representation of its fundamental group into PSL(2,C). The volume of \rho is defined by taking any \rho-equivariant map from the universal cover of W to H^3 and then by integrating the pull-back of the hyperbolic volume form on a fundamental domain. It turns out that such a volume does not depend on the choice of the equivariant map. Dunfield extended this construction to the case of a non-compact (cusped) manifold M, but he did not prove the volume is well-defined in all cases. We prove here that the volume of a representation is always well-defined and depends only on the representation. We show that this volume can be easily computed by straightening any ideal triangulation of M. We show that the volume of a representation is bounded from above by the relative simplicial volume of M. Finally, we prove a rigidity theorem for representations of the fundamental group of a hyperbolic manifold. Namely, we prove that if M is hyperbolic and vol(\rho)=vol(M) then \rho is discrete and faithful.

Affiliated with

Also associated with

No associations

Rating

Hyperbolic volume of representations of fundamental groups of cusped 3-manifolds does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Hyperbolic volume of representations of fundamental groups of cusped 3-manifolds, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Hyperbolic volume of representations of fundamental groups of cusped 3-manifolds will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-154139