Maximal surface group representations in isometry groups of classical Hermitian symmetric spaces

By Steven B. Bradlow, O. Garc'ia-Prada, Peter B. Gothen. 2005

129 citations recorded by the source corpus.

Higgs bundles and non-abelian Hodge theory provide holomorphic methods with which to study the moduli spaces of surface group representations in a reductive Lie group G. In this paper we survey the case in which G is the isometry group of a classical Hermitian symmetric space of non-compact type. Using Morse theory on the moduli spaces of Higgs bundles, we compute the number of connected components of the moduli space of representations with maximal Toledo invariant

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4850858c2dd5cb3447a780a767c17853d1f114ae

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

An invitation to higher Teichm\"uller theory (2018, methodology): “age. Remark 6. There are two types of Hermitian Lie groups, those of tube type and those not of tube-type. Maximal representations into Lie groups that are not of tube type satisfy a rigidity theorem [111, 67, 28, 21, 19]: The image of a maximal representation is always contained in the stabilizer i”
Topological invariants of Anosov representations (2009, methodology): “Where these methods have been applied to study representations into Lie groups of Hermitian type, see [19, 15, 14, 7, 8], the authors associate special vector bundles to the Higgs bundles, whose characteristic classes give additional invariants for maximal representations π1(Σ) → G; then they show t”
Cross ratios, surface groups, PSL(n,ℝ) and diffeomorphisms of the circle (2007, result): “In [22], cross ratios also appear in relation with maximal representations, as discussed in the work of Burger, Iozzi and Wienhard [8,9] and Bradlow, Garcia-Prada, Gothen, Mundet i Riera [6, 5 ,7,12,17].”
Higher Teichmüller Spaces: from SL(2,R) to other Lie groups (2010, methodology): “In many cases the exact number of connected components of the space of maximal representations has been computed using methods from the theory of Higgs bundles [50, 40, 39, 10, 11].”
The geometry of maximal representations of surface groups into SO0(2,n) (2017, background): “[BGPG06] The space of maximal SO0(2, 3)-representations decomposes as ⊔”