Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-25T17:24:44.609Z Has data issue: false hasContentIssue false
  • English
  • Français

Rigidity of diagonally embedded triangle groups

Part of: Special aspects of infinite or finite groups Lie groups

Published online by Cambridge University Press:  20 August 2020

Jean-Philippe Burelle
Jean-Philippe Burelle*
Affiliation:
Département de mathématiques, Université de Sherbrooke, Sherbrooke, QC J1K 2R1, Canada
*
e-mail: j-p.burelle@usherbrooke.ca
Get access Rights & Permissions [Opens in a new window]

Abstract

We show local rigidity of hyperbolic triangle groups generated by reflections in pairs of n-dimensional subspaces of $\mathbb {R}^{2n}$ obtained by composition of the geometric representation in $\mathsf {PGL}(2,\mathbb {R})$ with the diagonal embeddings into $\mathsf {PGL}(2n,\mathbb {R})$ and $\mathsf {PSp}^\pm (2n,\mathbb {R})$ .

Keywords

Triangles groupsrigidityreflectionscharacter varietiessymplectic group

MSC classification

Primary: 22E40: Discrete subgroups of Lie groups
Secondary: 20F55: Reflection and Coxeter groups
Type
Article
Information
Canadian Mathematical Bulletin , Volume 64 , Issue 3 , September 2021 , pp. 560 - 579
Check for updates
Copyright
© Canadian Mathematical Society 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alessandrini, D., Lee, G.-S., and Schaffhauser, F., Hitchin components for orbifolds. Preprint, 2018. arXiv:1811.05366 Google Scholar
Burger, M., Iozzi, A., and Wienhard, A., Surface group representations with maximal toledo invariant . Ann. of Math. 172(2010), 517–566. http://dx.doi.org/10.4007/annals.2010.172.517.CrossRefGoogle Scholar
Ghys, E. and Ranicki, A., Signatures in algebra, topology and dynamics . In: Six papers on signatures, braids and Seifert surfaces, Ensaios Mat., 30, Soc. Brasil Mat., Rio de Janeiro, 2016, pp. 1–173.Google Scholar
Guichard, O. and Wienhard, A., Topological invariants of Anosov representations . J. Topol. 3(2010), 578–642. http://dx.doi.org/10.1112/jtopol/jtq018.CrossRefGoogle Scholar
Guichard, O. and Wienhard, A., Anosov representations: domains of discontinuity and applications . Invent. Math. 190(2012), 357–438. http://dx.doi.org/10.1007/s00222-012-0382-7.CrossRefGoogle Scholar
Hoban, R., Local rigidity of triangle groups in Sp(4, R). Ph.D. thesis, University of Maryland, College Park, 2009. http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:3359280.Google Scholar
Labourie, F., Anosov flows, surface groups and curves in projective space . Invent. Math. 165(2006), 51–114. http://dx.doi.org/10.1007/s00222-005-0487-3.CrossRefGoogle Scholar
Long, D. and Thistlethwaite, M., The dimension of the hitchin component for triangle groups. Geom. Dedicata, to appear.Google Scholar
Vinberg, E. B., Discrete linear groups that are generated by reflections . Izv. Akad. Nauk SSSR Ser. Mat. 35(1971), 1072–1112.Google Scholar
Weir, E. A., The dimension of the restricted Hitchin component for triangle groups. PhD dissertation, University of Tennessee, Knoxville, 2018. https://trace.tennessee.edu/utk_graddiss/5089 Google Scholar
Wienhard, A., An invitation to higher Teichmüller theory. Preprint, 2018. arXiv:1803.06870 Google Scholar
Wonenburger, M. J., Simultaneous diagonalization of symmetric bilinear forms . J. Math. Mech. 15(1966), 617–622. http://dx.doi.org/10.1512/iumj.1966.15.15041.Google Scholar