Topological invariants of Anosov representations

By Olivier Y Guichard, Anna Wienhard. 2009

57 citations recorded by the source corpus.

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 components of the quotient by the mapping class group action. For specific symplectic maximal representations we compute the invariants explicitly. This allows us to construct nice model representations in all connected components. The construction of model representations is of particular interest for Sp (4, R), because in this case there are −1−χ(Σ) connected components in which all representations are Zariski dense and no model representations have been known so far. Finally, we use the model representations to draw conclusions about the holonomy of symplectic maximal representations.

Subfields: Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/aaa06f91be8851df1c0d31e9ecf95daea3186e31

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Citations in the corpus

Anosov representations: domains of discontinuity and applications (2011, background): “The quotient manifolds M = π1( )\ ρ will be investigated in more detail in [42], using results on topological invariants for maximal representations from [41].”
An invitation to higher Teichm\"uller theory (2018, methodology): “Such additional invariants have been defined on the one hand using methods from the theory of Higgs bundles [56, 19, 32, 9, 6] and on the other hand using the Anosov property of representations [63].”
Higher Teichmüller Spaces: from SL(2,R) to other Lie groups (2010, background): “2n,R) as the stabilizer of two transverse Lagrangian subspaces. The topological invariants of signiﬁcance are ﬁrst and second Stiefel–Whitney classes, as well as an Euler class if n= 2. Theorem 8.20 ([56]). The topological invariants distinguish the connected components of Hommax π1(S),Sp(2n,R) \Hom”