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

By S. Francaviglia. 2003

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Let W be a compact manifold and let ρ be a representation of its fundamental group into PSL(2, C). Then the volume of ρ is defined by taking any ρ-equivariant map from the universal cover W~ to ℍ 3 and then by integrating the pullback 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 noncompact (cusped) manifold M, but he did not prove that 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. Moreover, 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(ρ) = vol(M), then ρ is discrete and faithful.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a133c50c130748c5c9af29bba2c2ff71b0d057e8

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

A Dual Interpretation of the Gromov–Thurston Proof of Mostow Rigidity and Volume Rigidity for Representations of Hyperbolic Lattices (2012, background): “Vol(ρ)of a representation. Dunﬁeld introduced for this purpose th e notion of pseudodeveloping map and Francaviglia proved that the deﬁnitio n is independent of the choice of the pseudodeveloping map [18]. Then Francaviglia and Klaff [19] proved a “volume rigidity theorem” for representations ρ: π1(”
Gluing equations for PGL(n, ℂ)–representations of 3–manifolds (2012, background): “This is non-trivial even for n = 2, where it was first proved by Francaviglia [9].”