Surface group representations with maximal Toledo invariant

By M. Burger, A. Iozzi, Anna Wienhard. 2003

274 citations recorded by the source corpus.

We develop the theory of maximal representations of the fundamental group π 1 (Σ) of a compact connected oriented surface Σ (possibly with boundary) into Lie groups G of Hermitian type. For any homomorphism p: π 1 (Σ) → G, we define the Toledo invariant T(Σ, p), a numerical invariant which has both topological and analytical interpretations. We establish important properties of T(Σ, ρ), among which continuity, uniform boundedness on the representation variety, additivity under connected sum of surfaces and congruence relations mod 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. Moreover we establish properties of boundary maps associated to maximal representations which generalize naturally monotonicity properties of semiconjugations of the circle. We define a rotation number function for general locally compact groups and study it in detail for groups of Hermitian type. Properties of the rotation number, together with the existence of boundary maps, lead to additional invariants for maximal representations and show that the subset of maximal representations is always real semialgebraic.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d3142ba60cd42ad35fe0d295eba9c682750adec0

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Citations in the corpus

An invitation to higher Teichm\"uller theory (2018, methodology): “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 in G of a maximal subsymmetric space of tube type.”
Convex cocompactness in pseudo-Riemannian hyperbolic spaces (2017, background): “By [13,15], when p = m + 1 = 2 and is a closed surface group, the space Tρ0 is a full connected component ofHom( ,PO(2, q)), consistingof so-calledmaximal representations.”
Locally Homogeneous Geometric Manifolds (2010, background): “BurgerIozzi-Wienhard [28] showed that maximal representations are discrete embeddings, with reductive Zariski closures.”
Conjugation-invariant norms on groups of geometric origin (2007, background): “The only known to us proof of this innocently looking fact is surprisingly involved: it can be extracted from [7] (thanks to A. Iozzi and A. Wienhard for illuminating consultations).”
A Dual Interpretation of the Gromov–Thurston Proof of Mostow Rigidity and Volume Rigidity for Representations of Hyperbolic Lattices (2012, methodology): “The aim of this paper is to give a complete proof of volume rigi dity from the point of view of bounded cohomology, implementing a strateg y first described in [24] and used in the work on maximal representations of surface g roups [12, 13], as well as in the proof of Mostow Rigidity in dimension 3 ”