Groupes triangulaires lagrangiens en géométrie hyperbolique complexe

Pierre Will[1]

  • [1] Université Pierre et Marie Curie Institut de Mathématiques 4 place Jussieu 75252 Paris cedex 05 (France)

Séminaire de théorie spectrale et géométrie (2006-2007)

  • Volume: 25, page 189-209
  • ISSN: 1624-5458

Abstract

top
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.

How to cite

top

Will, Pierre. "Groupes triangulaires lagrangiens en géométrie hyperbolique complexe." Séminaire de théorie spectrale et géométrie 25 (2006-2007): 189-209. <http://eudml.org/doc/11225>.

@article{Will2006-2007,
abstract = {Nous présentons quelques résultats au sujet des groupes engendrés par trois involutions antiholomorphes dans le cadre du plan hyperbolique complexe $\mathbf\{H\}^\{2\}_\{\mathbb\{C\}\}$.},
affiliation = {Université Pierre et Marie Curie Institut de Mathématiques 4 place Jussieu 75252 Paris cedex 05 (France)},
author = {Will, Pierre},
journal = {Séminaire de théorie spectrale et géométrie},
language = {fre},
pages = {189-209},
publisher = {Institut Fourier},
title = {Groupes triangulaires lagrangiens en géométrie hyperbolique complexe},
url = {http://eudml.org/doc/11225},
volume = {25},
year = {2006-2007},
}

TY - JOUR
AU - Will, Pierre
TI - Groupes triangulaires lagrangiens en géométrie hyperbolique complexe
JO - Séminaire de théorie spectrale et géométrie
PY - 2006-2007
PB - Institut Fourier
VL - 25
SP - 189
EP - 209
AB - Nous présentons quelques résultats au sujet des groupes engendrés par trois involutions antiholomorphes dans le cadre du plan hyperbolique complexe $\mathbf{H}^{2}_{\mathbb{C}}$.
LA - fre
UR - http://eudml.org/doc/11225
ER -

References

top
  1. M. Deraux, E. Falbel, and J. Paupert. New constructions of fundamental polyhedra in complex hyperbolic space. Acta Math., 194 :155–201, 2005. Zbl1113.22010MR2231340
  2. E. Falbel and P.V. Koseleff. A circle of modular groups in P U ( 2 , 1 ) . Math. Res. Let., 9 :379–391, 2002. Zbl1008.20038MR1909651
  3. E. Falbel and P.V. Koseleff. Rigidity and flexibility of triangle groups in complex hyperbolic geometry. Topology, 41, 2002. Zbl1005.32018MR1905838
  4. E. Falbel and J. Parker. The moduli space of the modular group. Inv. Math., 152, 2003. Zbl1160.32306MR1965360
  5. E. Falbel and V. Zocca. A P oincaré polyhedron theorem for complex hyperbolic geometry. J. reine angew. Math., 516 :133–158, 1999. Zbl0944.53042MR1724618
  6. W. Goldman. Complex Hyperbolic Geometry. Oxford University Press, Oxford, 1999. Zbl0939.32024MR1695450
  7. W. Goldman and J. Parker. Complex hyperbolic ideal triangle groups. Journal für dir reine und angewandte Math., 425 :71–86, 1992. Zbl0739.53055MR1151314
  8. N. Gusevskii and J.R. Parker. Complex hyperbolic quasi-fuchsian groups and T oledo’s invariant. Geom. Ded., 97 :151–185, 2003. Zbl1042.57023
  9. G. D. Mostow. A remarkable class of polyhedra in complex hyperbolic space. Pac. J. Math., 86 :171–276, 1980. Zbl0456.22012MR586876
  10. J. Parker and I. Platis. Open sets of maximal dimension in complex hyperbolic quasi-fuchsian space. J. Diff. Geom, 73 :319–350, 2006. Zbl1100.30037MR2226956
  11. A. Pratoussevitch. Traces in complex hyperbolic triangle groups. Geometriae Dedicata, 111 :159–185, 2005. Zbl1115.32015MR2155180
  12. H. Sandler. Traces on S U ( 2 , 1 ) and complex hyperbolic ideal triangle groups. Algebras groups and geometries, 12 :139–156, 1995. Zbl0910.20032MR1325978
  13. R. E. Schwartz. Degenerating the complex hyperbolic ideal triangle groups. Acta Math., 186 :105–154, 2001. Zbl0998.53050MR1828374
  14. R. E. Schwartz. Ideal triangle groups, dented tori, and numerical analysis. Ann. of Math. (2), 153 :533–598, 2001. Zbl1055.20040MR1836282
  15. R. E. Schwartz. Complex hyperbolic triangle groups. Proc. Int. Math. Cong., 1 :339–350, 2002. Zbl1022.53034MR1957045
  16. R. E. Schwartz. A better proof of the G oldman- P arker conjecture. Geometry and Topology, 9, 2005. Zbl1098.20034MR2175152
  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 P U (2,1). Preprint disponible sur www.math.jussieu.fr/ will 
  19. P. Will. The punctured torus and L agrangian triangle groups in P U ( 2 , 1 ) . J. reine angew. Math., 602 :95–121, 2007. Zbl1117.32010MR2300453

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.