Maximal Representations of Surface Groups: Symplectic Anosov Structures

By M. Burger, A. Iozzi, F. Labourie, Anna Wienhard. 2005

116 citations recorded by the source corpus.

Let G be a connected semisimple Lie group such that the associ- ated symmetric space X is Hermitian and let Γg be the fundamental group of a compact orientable surface of genus g ≥ 2. We survey the study of maximal representations of Γg into G, that is the subset of Hom(Γg ,G ) characterized by the maximality of the Toledo invariant ((17) and (15)). Then we concen- trate on the particular case G =S p(2n, R), and we show that if ρ is any maximal representation then the image ρ(Γg) is a discrete, faithful realiza- tions of Γg as a Kleinian group of complex motions in X with an associated Anosov system, and whose limit set in an appropriate compactification of X is a rectifiable circle.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/602ac5527981e512223d130ad268ea797a67ee80

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Citations in the corpus

Higher Teichmüller Spaces: from SL(2,R) to other Lie groups (2010, background): “(For an illustration of this strategy for maximal representations into the symplectic group we refer the reader to [14].”
The geometry of maximal representations of surface groups into SO0(2,n) (2017, background): “groups SO0(2,n + 1), n ≥ 1 (see Remark 1.10). Geometrization of maximal representations. Hitchin representationsintosplit real Lie groups [Lab06] and maximal representations into Hermitian Lie groups [BILW05] are two important families of Anosov representations. One nice feature MAXIMAL REPRESENTATI”
Topological invariants of Anosov representations (2009, background): “[9, 10] A maximal representation ρ : π1(Σ) → G is an Anosov representation.”
Convex Foliated Projective Structures and the Hitchin Component for PSL(4,R) (2007, background): “Moreover, for (t+, t0, t−) the three Lagrangians ξ (2)(t+), ξ (2)(t0), ξ (2)(t−) are pairwise transverse and define a complex structure J(t+,t0,t−) on R 4 = ξ(2)(t−)⊕ξ (2)(t+) such that the symmetric bilinear form ω(·, J(t+ ,t0,t+)·) is positive definite (this is true in greater generality for maxim”
Surface group representations with maximal Toledo invariant (2003, methodology): “In [12] we have shown that if S D nD is a closed hyperbolic surface and W ! Sp.”
Locally Homogeneous Geometric Manifolds (2010, background): “tly been studied by Barbot [6, 7]. The properness of the action of Mod(Σ) on F(Σ) is generally attributed to Fricke. Many cases are known of components of deformation spaces when Mod(Σ) acts properly [94, 171, 27]. In many of these cases, these components consist of holonomy representations of unifo”
Bounded differential forms, generalized Milnor–Wood inequality and an application to deformation rigidity (2007, unknown): “subject of much research over the years ([30], [31], [29], [34], [42], [23], [32], [38], [6], [18], [41], [ 10 ], [15], [16], [17]).”
An invitation to higher Teichm\"uller theory (2018, background): “Representations in the Hitchin component and maximal representations were the first examples of Anosov representations [75, 25, 64, 27].”
Maximal surface group representations in isometry groups of classical Hermitian symmetric spaces (2005, unknown): “More recently, Labourie [26] has shown for G = SL(n, R) that these representations are discrete and faithful and are related to certain Anosov geometric structures, and analogous results have been obtained for G = Sp(2n, R) by Burger et al. [ 6 ].”
Anosov representations: domains of discontinuity and applications (2011, background): “A corollary of this and [18] is (see also Remark 6.”