Geodesic
Surface group representations with maximal Toledo invariant
Marc Burger1 ; Alessandra Iozzi2 ; Anna Wienhard3
1 FIM, ETH Zentrum, Rämistrasse 101, CH-8092 Zürich, Switzerland
2 D-Math, ETH Zentrum, Rämistrasse 101, CH-8092 Zürich, Switzerland
3 Department of Mathematics, >Princeton University, Fine Hall - Washington Road, Princeton, NJ 08540, United States
Annals of mathematics, Tome 172 (2010) no. 1, pp. 517-566.

Voir la notice de l'article provenant de la source Annals of Mathematics website

We develop the theory of maximal representations of the fundamental group $\pi_1(\Sigma)$ of a compact connected oriented surface $\Sigma$ (possibly with boundary) into Lie groups $G$ of Hermitian type. For any homomorphism $\rho:\pi_1(\Sigma)\to G$, we define the Toledo invariant $\operatorname{T}(\Sigma,\rho)$, a numerical invariant which has both topological and analytical interpretations. We establish important properties of $\operatorname{T}(\Sigma,\rho)$, among which continuity, uniform boundedness on the representation variety, additivity under connected sum of surfaces and congruence relations mod $\mathbb{Z}$. We thus obtain information about the representation variety as well as striking geometric properties of maximal representations, that is representations whose Toledo invariant achieves the maximum value.
DOI : 10.4007/annals.2010.172.517

Marc Burger 1 ; Alessandra Iozzi 2 ; Anna Wienhard 3

1 FIM, ETH Zentrum, Rämistrasse 101, CH-8092 Zürich, Switzerland
2 D-Math, ETH Zentrum, Rämistrasse 101, CH-8092 Zürich, Switzerland
3 Department of Mathematics, >Princeton University, Fine Hall - Washington Road, Princeton, NJ 08540, United States 
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Marc Burger; Alessandra Iozzi; Anna Wienhard. Surface group representations with maximal Toledo invariant. Annals of mathematics, Tome 172 (2010) no. 1, pp. 517-566. doi : 10.4007/annals.2010.172.517. http://geodesic.mathdoc.fr/articles/10.4007/annals.2010.172.517/

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