Topological invariants of Anosov representations

@article{Guichard2009TopologicalIO,
  title={Topological invariants of Anosov representations},
  author={Olivier Y Guichard and Anna Wienhard},
  journal={Journal of Topology},
  year={2009},
  volume={3}
}
We define new topological invariants for Anosov representations and study them in detail for maximal representations of the fundamental group of a closed oriented surface Σ into the symplectic group Sp (2n, R). In particular we show that the invariants distinguish connected components of the space of symplectic maximal representations other than Hitchin components. Since the invariants behave naturally with respect to the action of the mapping class group of Σ, we obtain from this the number of… 

Anosov representations: domains of discontinuity and applications

The notion of Anosov representations has been introduced by Labourie in his study of the Hitchin component for SL(n,R). Subsequently, Anosov representations have been studied mainly for surface

Anosov representations: domains of discontinuity and applications

The notion of Anosov representations has been introduced by Labourie in his study of the Hitchin component for SL(n,R). Subsequently, Anosov representations have been studied mainly for surface

Topological invariants for Anosov representations

Our goal in these notes will be to define topological invariants for Anosov representations into Sp(2n,R), concentrating particularly on the Sp(4,R) case, which allow us to distinguish connected

Parametrizing spaces of positive representations

. Using Lusztig’s total positivity in split real Lie groups V. Fock and A. Goncharov have introduced spaces of positive (framed) representations. For general semisimple Lie groups a generalization of

A note on Reidemeister Torsion of G-Anosov Representations

This article considers G-Anosov representations of a fixed closed oriented Riemann surface Σ of genus at least 2. Here, G is the Lie group PSp(2n,R), PSO(n, n) or PSO(n, n + 1). It proves that

The geometry of maximal components of the PSp(4, ℝ) character variety

In this paper we describe the space of maximal components of the character variety of surface group representations into PSp(4,R) and Sp(4,R). For every rank 2 real Lie group of Hermitian type, we

Higgs bundles for the non-compact dual of the special orthogonal group

Higgs bundles over a closed orientable surface can be defined for any real reductive Lie group $$G$$G. In this paper we examine the case $$G=\mathrm {SO}^*(2n)$$G=SO∗(2n). We describe a rigidity

Lorentzian Kleinian Groups

Classical Kleinian groups are discrete subgroups of isometries of H n. The well-known theory of Kleinian groups starts with the definition of their associated limit set in the boundary of H n , and

SO(p,q)-Higgs bundles and higher Teichm\"uller components

Some connected components of a moduli space are mundane in the sense that they are distinguished only by obvious topological invariants or have no special characteristics. Others are more alluring

On partial abelianization of framed local systems

. D. Gaiotto, G. W. Moore and A. Neitzke introduced spectral networks to understand the framed G -local systems over punctured surfaces for G a split Lie group via a procedure called abelianization.
...

References

SHOWING 1-10 OF 45 REFERENCES

Surface group representations with maximal Toledo invariant

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

Maximal Representations of Surface Groups: Symplectic Anosov Structures

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

Moduli spaces of local systems and higher Teichmüller theory

Let G be a split semisimple algebraic group over Q with trivial center. Let S be a compact oriented surface, with or without boundary. We define positive representations of the fundamental group of S

Surface group representations and U(p, q)-Higgs bundles

Using the L2 norm of the Higgs field as a Morse function, we study the moduli spaces of U(p, q)-Higgs bundles over a Riemann surface. We require that the genus of the surface be at least two, but

Components of spaces of representations and stable triples

Anosov flows, surface groups and curves in projective space

Note that in [10], W. Goldman gives a complete description of these connected components in the case of finite covers of PSL(2,R). In the case of PSL(2,R), two homeomorphic components, called

Anosov AdS representations are quasi-Fuchsian

Let Gamma be a cocompact lattice in SO(1,n). A representation rho: Gamma \to SO(2,n) is quasi-Fuchsian if it is faithfull, discrete, and preserves an acausal subset in the boundary of anti-de Sitter

Quasi-Fuchsian AdS representations are Anosov

In a recent paper, Q. M\'erigot proved that representations in SO(2,n) of uniform lattices of SO(1,n) which are Anosov in the sense of Labourie are quasi-Fuchsian, i.e. are faithfull, discrete, and

Mapping Class Groups

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

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