## Geometry & Topology

### Quakebend deformations in complex hyperbolic quasi-Fuchsian space

Ioannis D Platis

#### Abstract

We study quakebend deformations in complex hyperbolic quasi-Fuchsian space $Qℂ(Σ)$ of a closed surface $Σ$ of genus $g>1$, that is the space of discrete, faithful, totally loxodromic and geometrically finite representations of the fundamental group of $Σ$ into the group of isometries of complex hyperbolic space. Emanating from an $ℝ$–Fuchsian point $ρ∈Qℂ(Σ)$, we construct curves associated to complex hyperbolic quakebending of $ρ$ and we prove that we may always find an open neighborhood $U(ρ)$ of $ρ$ in $Qℂ(Σ)$ containing pieces of such curves. Moreover, we present generalisations of the well known Wolpert–Kerckhoff formulae for the derivatives of geodesic length function in Teichmüller space.

#### Article information

Source
Geom. Topol., Volume 12, Number 1 (2008), 431-459.

Dates
Accepted: 6 December 2007
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513800024

Digital Object Identifier
doi:10.2140/gt.2008.12.431

Mathematical Reviews number (MathSciNet)
MR2390350

Zentralblatt MATH identifier
1153.30038

Keywords
complex hyperbolic bending

#### Citation

Platis, Ioannis D. Quakebend deformations in complex hyperbolic quasi-Fuchsian space. Geom. Topol. 12 (2008), no. 1, 431--459. doi:10.2140/gt.2008.12.431. https://projecteuclid.org/euclid.gt/1513800024

#### References

• B Apanasov, Bending deformations of complex hyperbolic surfaces, J. Reine Angew. Math. 492 (1997) 75–91
• D B A Epstein, A Marden, Convex hulls in hyperbolic space, a theorem of Sullivan, and measured pleated surfaces, from: “Analytical and geometric aspects of hyperbolic space (Coventry/Durham, 1984)”, London Math. Soc. Lecture Note Ser. 111, Cambridge Univ. Press, Cambridge (1987) 113–253
• E Falbel, R A Wentworth, Eigenvalues of products of unitary matrices and Lagrangian involutions, Topology 45 (2006) 65–99
• W M Goldman, Complex hyperbolic geometry, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press (1999)
• W M Goldman, M Kapovich, B Leeb, Complex hyperbolic manifolds homotopy equivalent to a Riemann surface, Comm. Anal. Geom. 9 (2001) 61–95
• S P Kerckhoff, The Nielsen realization problem, Ann. of Math. $(2)$ 117 (1983) 235–265
• C Kourouniotis, Deformations of hyperbolic structures, Math. Proc. Cambridge Philos. Soc. 98 (1985) 247–261
• C Kourouniotis, Bending in the space of quasi-Fuchsian structures, Glasgow Math. J. 33 (1991) 41–49
• C Kourouniotis, The geometry of bending quasi-Fuchsian groups, from: “Discrete groups and geometry (Birmingham, 1991)”, London Math. Soc. Lecture Note Ser. 173, Cambridge Univ. Press, Cambridge (1992) 148–164
• J R Parker, I D Platis, Open sets of maximal dimension in complex hyperbolic quasi-Fuchsian space, J. Differential Geom. 73 (2006) 319–350
• J R Parker, I D Platis, Complex hyperbolic Fenchel–Nielsen coordinates, Topology (2007) To appear.
• J R Parker, C Series, Bending formulae for convex hull boundaries, J. Anal. Math. 67 (1995) 165–198
• I D Platis, Complex symplectic geometry of quasi-Fuchsian space, Geom. Dedicata 87 (2001) 17–34
• C Series, An extension of Wolpert's derivative formula, Pacific J. Math. 197 (2001) 223–239
• W P Thurston, Three-dimensional geometry and topology. Vol. 1, Princeton Mathematical Series 35, Princeton University Press, Princeton, NJ (1997) Edited by Silvio Levy
• D Toledo, Representations of surface groups in complex hyperbolic space, J. Differential Geom. 29 (1989) 125–133
• P Will, Lagrangian decomposability of some two-generator subgroups of $\rm PU(2,1)$, C. R. Math. Acad. Sci. Paris 340 (2005) 353–358
• S Wolpert, The Fenchel–Nielsen deformation, Ann. of Math. $(2)$ 115 (1982) 501–528
• S Wolpert, On the symplectic geometry of deformations of a hyperbolic surface, Ann. of Math. $(2)$ 117 (1983) 207–234