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On deformation spaces of nonuniform hyperbolic lattices

Published online by Cambridge University Press:  03 May 2016

SUNGWOON KIM  and
INKANG KIM
SUNGWOON KIM
Affiliation:
Department of Mathematics, Jeju National University, Jeju, Republic of Korea. e-mail: sungwoon@jejunu.ac.kr
INKANG KIM
Affiliation:
School of Mathematics, Korea Institute for Advanced Study, Seoul, Republic of Korea. e-mail: inkang@kias.re.kr
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Abstract

Let Γ be a nonuniform lattice acting on the real hyperbolic n-space. We show that in dimension greater than or equal to 4, the volume of a representation is constant on each connected component of the representation variety of Γ in SO(n, 1). Furthermore, in dimensions 2 and 3, there is a semialgebraic subset of the representation variety such that the volume of a representation is constant on connected components of the semialgebraic subset. Combining our approach with the main result of [2] gives a new proof of the local rigidity theorem for nonuniform hyperbolic lattices and the analogue of Soma's theorem, which shows that the number of orientable hyperbolic manifolds dominated by a closed, connected, orientable 3-manifold is finite, for noncompact 3-manifolds.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 161 , Issue 2 , September 2016 , pp. 283 - 303
Copyright
Copyright © Cambridge Philosophical Society 2016 

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References

REFERENCES

[1] Bucher, M., Burger, M., Frigerio, R., Iozzi, A., Pagliantini, C. and Pozzetti, M. B. Isometric embeddings in bounded cohomology. J. Topol. Anal. 6 (2014), no. 1, 1–25.Google Scholar
[2] Bucher, M., Burger, M. and Iozzi, A. A dual interpretation of the Gromov–Thurston proof of Mostow rigidity and volume rigidity for representations of hyperbolic lattices. In: Trends in Harmonic Analysis, pp. 47–76, Springer INdAM Ser., 3 (Springer-Verlag, Berlin, 2013).Google Scholar
[3] Bucher, M., Burger, M. and Iozzi, A. Integrality of volumes of representations. arXiv:1407.0562v1 [math.GT].Google Scholar
[4] Besson, G., Courtois, G. and Gallot, S. Inégalités de Milnor–Wood géométriques. Comment. Math. Helv. 82 (2007), no. 4, 753–803.Google Scholar
[5] Burger, M., Iozzi, A. and Wienhard, A. Surface group representations with maximal Toledo invariant. Ann. of Math. (2) 172 (2010), no. 1, 517–566.Google Scholar
[6] Bucher, M., Kim, I. and Kim, S. Proportionality principle for the simplicial volume of families of ℚ-rank 1 locally symmetric spaces. Math. Z. 276 (2014), no. 1-2, 153–172.CrossRefGoogle Scholar
[7] Calabi, E. On compact Riemannian manifolds with constant curvature. I. In: Proc. Sympos. Pure Math. III (1961), pp. 155–180 (AMS, Providence, RI).Google Scholar
[8] Dunfield, N. M. Cyclic surgery, degrees of maps of character curves, and volume rigidity for hyperbolic manifolds. Invent. Math. 136 (1999), no. 3, 623–657.Google Scholar
[9] Francaviglia, S. Hyperbolic volume of representations of fundamental groups of cusped 3-manifolds. Int. Math. Res. Not. (2004), no. 9, 425–459.Google Scholar
[10] Francaviglia, S. and Klaff, B. Maximal volume representation are Fuchsian. Geom. Dedicata 117 (2006), 111–124.Google Scholar
[11] Guichardet, A. Cohomologie des Groupes Topologiques et des Algèbres de Lie (CEDIC, Paris, 1980).Google Scholar
[12] Goldman, W. M. Topological components of spaces of representations. Invent. Math. 93 (1988), no. 3, 557–607.Google Scholar
[13] Hodgson, C. Degeneration and regeneration of geometric structures on three-manifolds. PhD. thesis (Princeton University, 1986).Google Scholar
[14] Kapovich, M. On dynamics of pseudo-Anosov homeomorphisms on representation varieties of surface groups. Ann. Acad. Sci. Fenn. Math. 23 (1998), no. 1, 83–100.Google Scholar
[15] Kim, S. On the equivalence of the definitions of volume of representations. Pacific J. Math. 280 (2016), no. 1, 51–68.CrossRefGoogle Scholar
[16] Kim, S. and Kim, I. Volume invariant and maximal representations of discrete subgroups of Lie groups. Math. Z. 276 (2014), no. 3-4, 1189–1213.CrossRefGoogle Scholar
[17] Kim, S. and Kuessner, T. Simplicial volume of compact manifolds with amenable boundary. J. Topol. Anal. 7 (2015), no. 1, 23–46.Google Scholar
[18] Kneser, H. Der Simplexinhalt in der nichteuklidischen Geometrie. Deutsche Math. 1 (1936), 337–340.Google Scholar
[19] Löh, C. ℓ1-Homology and Simplicial Volume. PhD thesis. (WWU Münster, 2007), http://www.mathematik.uni-regensburg.de/loeh/theses/thesis.pdf.Google Scholar
[20] Milnor, J. The Schläfli differential equality. In Collected Papers, vol. 1. (Publish or Perish, 1994).Google Scholar
[21] Monod, N. Continuous Bounded Cohomology of Locally Compact Groups. Lecture Notes in Math. 1758 (Springer-Verlag, Berlin, 2001).CrossRefGoogle Scholar
[22] Moore, C. C. Amenable subgroups of semisimple groups and proximal flows. Israel J. Math. 34 (1979), no. 1-2, 121–138.CrossRefGoogle Scholar
[23] Munkres, J. R. Elementary Differential Topology (Princeton University Press, Princeton, NJ, 1966).Google Scholar
[24] Neumann, W. D. and Yang, J. Bloch invariants of hyperbolic 3-manifolds. Duke Math. J. 96 (1999), no. 1, 29–59.Google Scholar
[25] Reznikov, A. Rationality of secondary classes. J. Differential Geom. 43 (1996), no. 3, 674–692.Google Scholar
[26] Soma, T. Non-Zero degree maps to hyperbolic 3-manifolds. J. Differential Geom. 49 (1998), no. 3, 517–546.Google Scholar
[27] Thurston, W. Geometry and Topology of 3-Manifolds. Lecture Notes, Princeton, (1978). http://library.msri.org/books/gt3m.Google Scholar
[28] Toledo, D. Representations of surface groups in complex hyperbolic space. J. Differential Geom. 29 (1989), no. 1, 125–133.Google Scholar
[29] Vinberg, E. B. editor. Geometry II, Geometry of Spaces of Constant Curvature Encyclopaedia of Mathematical Sciences vol. 29 (Springer, 1993).Google Scholar
[30] Weil, A. On discrete subgroups of Lie groups. Ann. of Math. (2) 72 (1960), 369–384.Google Scholar
[31] Weil, A. On discrete subgroups of Lie groups II. Ann. of Math. (2) 75 (1962), 578–602.Google Scholar
[32] Zickert, C. K. The volume and Chern–Simon invariant of a representation. Duke Math. J. 150 (2009), no. 3, 489–532.Google Scholar