Center for diffusion of mathematic journals

 
 
 
 

Séminaire de Théorie spectrale et géométrie (Grenoble)

Table of contents for this volume | Previous article | Next article
Pierre Will
Groupes triangulaires lagrangiens en géométrie hyperbolique complexe
Séminaire de Théorie spectrale et géométrie (Grenoble), 25 (2006-2007), p. 189-209, doi: 10.5802/tsg.256
Article PDF | Reviews MR 2478817 | Zbl 1159.32007
Class. Math.: 32G15, 32M15, 32Q45, 57S30

Résumé - Abstract

There is still much to learn about discrete subgroups of PU($n$,1), the group of holomorphic isometries of the complex hyperbolic n-space. Given a finitely generated group G, describing the discrete and faithful representation of G into PU($n$,1) is a difficult task, which has been carried out in only few cases. In this note we expose some results about Lagrangian triangle groups in the frame of PU(2,1). These groups are generated by three antiholomorphic isometric involutions. They are connected to many of the known examples of discrete subgroups of PU(2,1). The main result stated here is the existence of a one parameter family of embeddings of the Teichmueller space of the once punctured torus into the PU(2,1)-representation variety of the free group of rank two. These embeddings are described using Lagrangian triangle groups.

Bibliography

[1] M. Deraux, E. Falbel, and J. Paupert. New constructions of fundamental polyhedra in complex hyperbolic space. Acta Math., 194 :155–201, 2005.  MR 2231340 |  Zbl 1113.22010
[2] E. Falbel and P.V. Koseleff. A circle of modular groups in $\mbox {PU(2,1)}$. Math. Res. Let., 9 :379–391, 2002.  MR 1909651 |  Zbl 1008.20038
[3] E. Falbel and P.V. Koseleff. Rigidity and flexibility of triangle groups in complex hyperbolic geometry. Topology, 41, 2002.  MR 1905838 |  Zbl 1005.32018
[4] E. Falbel and J. Parker. The moduli space of the modular group. Inv. Math., 152, 2003.  MR 1965360 |  Zbl pre01947253
[5] E. Falbel and V. Zocca. A $\mbox {P}$oincaré polyhedron theorem for complex hyperbolic geometry. J. reine angew. Math., 516 :133–158, 1999.  MR 1724618 |  Zbl 0944.53042
[6] W. Goldman. Complex Hyperbolic Geometry. Oxford University Press, Oxford, 1999.  MR 1695450 |  Zbl 0939.32024
[7] W. Goldman and J. Parker. Complex hyperbolic ideal triangle groups. Journal für dir reine und angewandte Math., 425 :71–86, 1992.  MR 1151314 |  Zbl 0739.53055
[8] N. Gusevskii and J.R. Parker. Complex hyperbolic quasi-fuchsian groups and $\mbox {T}$oledo’s invariant. Geom. Ded., 97 :151–185, 2003.  Zbl 1042.57023
[9] G. D. Mostow. A remarkable class of polyhedra in complex hyperbolic space. Pac. J. Math., 86 :171–276, 1980. Article |  MR 586876 |  Zbl 0456.22012
[10] J. Parker and I. Platis. Open sets of maximal dimension in complex hyperbolic quasi-fuchsian space. J. Diff. Geom, 73 :319–350, 2006. Article |  MR 2226956 |  Zbl 1100.30037
[11] A. Pratoussevitch. Traces in complex hyperbolic triangle groups. Geometriae Dedicata, 111 :159–185, 2005.  MR 2155180 |  Zbl 1115.32015
[12] H. Sandler. Traces on ${ \mbox {SU}(2,1)}$ and complex hyperbolic ideal triangle groups. Algebras groups and geometries, 12 :139–156, 1995.  MR 1325978 |  Zbl 0910.20032
[13] R. E. Schwartz. Degenerating the complex hyperbolic ideal triangle groups. Acta Math., 186 :105–154, 2001.  MR 1828374 |  Zbl 0998.53050
[14] R. E. Schwartz. Ideal triangle groups, dented tori, and numerical analysis. Ann. of Math. (2), 153 :533–598, 2001.  MR 1836282 |  Zbl 1055.20040
[15] R. E. Schwartz. Complex hyperbolic triangle groups. Proc. Int. Math. Cong., 1 :339–350, 2002.  MR 1957045 |  Zbl 1022.53034
[16] R. E. Schwartz. A better proof of the $\mbox {G}$oldman-$\mbox {P}$arker conjecture. Geometry and Topology, 9, 2005.  MR 2175152 |  Zbl 1098.20034
[17] P. Will. Groupes libres, groupes triangulaires et tore épointé dans PU(2,1). Thèse de l’université Paris VI.
[18] P. Will. Traces, cross-ratios and 2-generator subgroups of $\mbox {PU}$(2,1). Preprint disponible sur www.math.jussieu.fr/ will
[19] P. Will. The punctured torus and $\mbox {L}$agrangian triangle groups in $\mbox {PU(2,1)}$. J. reine angew. Math., 602 :95–121, 2007.  MR 2300453 |  Zbl 1117.32010
Copyright Cellule MathDoc 2019 | Credit | Site Map