Abstract
We prove a volume-rigidity theorem for Fuchsian representations of fundamental groups of hyperbolic k-manifolds into Isom \(\mathbb{H}^n\). Namely, we show that if M is a complete hyperbolic k-manifold with finite volume, then the volume of any representation of π1(M) into isom \(\mathbb{H}^n\), 3 ≤ k ≤ n, is less than the volume of M, and the volume is maximal if and only if the representation is discrete, faithful and ‘k-Fuchsian’
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Benedetti R. and Petronio C. (1992). Lectures on Hyperbolic Geometry. Universitext, Springer-Verlag, Berlin
Besson G., Courtois G. and Gallot S. (1995). Entropies et rigidités des espaces localement symétriques de courbure strictement négative. Geom. Funct. Anal. 5(5):731–799
Besson G., Courtois G. and Gallot S. (1996). Minimal entropy and Mostow’s rigidity theorems. Ergodic Theory Dynam. Systems 16(4):623–649
Besson G., Courtois G. and Gallot S. (1999). Lemme de Schwarz réel et applications géométriques. Acta Math. 183(2):145–169
Besson, G., Courtois, G. and Gallot, S.: Inégalité de milnor-wood géométriques, 2004. In preparation.
Dunfield N.M. (1999). Cyclic surgery, degrees of maps of character curves, and volume rigidity for hyperbolic manifolds. Invent. Math. 136(3):623–657
Francaviglia, S.: Constructing equivariant maps for representations, Preprint DMA, available version arXiv:math.GT/0405028.
Francaviglia S. (2004). Hyperbolic volume of representations of fundamental groups of cusped 3-manifolds. Int. Math. Res. Not. 9:425–459
Gallot S. Hulin D. and Lafontaine J. (1990). Riemannian Geometry, 2nd edn, Universitext, Springer-Verlag, Berlin
Goldman W.M. (1982). Characteristic classes and representations of discrete subgroups of Lie groups. Bull. Amer. Math. Soc. (N.S.) 6(1):91–94
Klaff, B.: Boundary slopes of knots in closed 3-manifolds with cyclic fundamental group, PhD thesis, University Illinois-Chicago, 2003.
Thurston, W. P.: The geometry and topology of 3-manifolds, Mimeographed notes, Princeton University Mathematics Department, 1979.
Author information
Authors and Affiliations
Corresponding author
Additional information
Stefano Francaviglia: Supported by an INdAM and a Marie Curie Intra European fellowship
Ben Klaff: Supported by a CIRGET fellowship and by the Chaire de Recherche du Canada en algèbre, combinatoire et informatique mathématique de l’UQAM.
Rights and permissions
About this article
Cite this article
Francaviglia, S., Klaff, B. Maximal Volume Representations are Fuchsian. Geom Dedicata 117, 111–124 (2006). https://doi.org/10.1007/s10711-005-9033-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10711-005-9033-0