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Maximal representations of surface groups: symplectic Anosov structures. (English) Zbl 1157.53025

This paper surveys the theory of maximal representations of a closed surface group into the automorphism group \(G\) of a Hermitian symmetric space \(\mathcal X\) of \(\mathbb R\)-rank \(r\). Specifically, if \(\Gamma_g\) is the fundamental group of a closed oriented surface \(\Sigma_g\) of genus \(g > 1\), Toledo constructed an invariant \(T_\rho\) of representations \(\rho: \Gamma_g \to G\) which is locally constant on \(\text{Hom}(\Gamma_g , G)\) and satisfies an inequality
\[ |T_\rho|\leq r|\chi(\Sigma_g)| \]
(Domic-Toledo). The representation \(\rho\) is maximal if equality holds.
In his 1980 doctoral thesis the reviewer proved that when \(\mathcal X\) is the hyperbolic plane, maximal representations are precisely embeddings of \(\Gamma_g\) as a discrete subgroup of \(G \cong\text{PSL}(2,\mathbb R)\cong \text{PU}(1, 1)\). In this case \(T_\rho\) is the Euler class of the associated flat oriented \(\mathbb R\text{P}^1\)-bundle over \(\Sigma_g\) and the above inequality is due to Milnor and Wood.
Thus equivalence classes of maximal representations comprise the Teichmüller space of \(\Sigma_g\) (since the uniformization theorem identifies marked complex structures on \(\Sigma_g\) with marked hyperbolic structures on \(\Sigma_g\)).
The maximality of \(\rho\) may be characterized in many other equivalent ways: for example, that \(\rho\) defines a quasi-isometric embedding of the Cayley graph of \(\Gamma_g\) into \(\mathcal X\). Another equivalent condition is that for every \(\gamma\in\Gamma_g\setminus\{1\}\), the image \(\rho(\gamma)\) is a nontrivial hyperbolic element of \(G\).
The present paper discusses many remarkable properties of maximal representations for other groups \(G\). When \(G\cong\text{PU}(n, 1)\), so \(\mathcal X\) is complex hyperbolic \(n\)-space, then Toledo proved every maximal representation preserves a totally geodesic holomorphic curve in \(\mathcal X\). However, for \(G = \text{PU}(p, q)\), where \(p\geq q\), Bradlow, Garcia-Prada and Gother proved every reductive representation can be conjugated to the image of \(\text{P} (\text{U}(p - q) \times \text{U}(q, q))\). Conversely Burger, Iozzi and Wienhard constructed maximal representations which are Zariski dense in \(\text{PU}(q, q)\). In general a Hermitian symmetric space \(\mathcal X\) contains a maximal totally geodesic Hermitian symmetric space of tube type and maximal representations necessarily preserve this subspace. Furthermore Burger-Iozzi-Wienhard proved that every maximal representation is reductive.
The methods of Bradlow, Garcia-Prada and Gothern are completely different methods. Employing Morse theory on moduli spaces of Higgs bundles, they start from a Riemann surface \(M\approx \Sigma_g\), and identify representations of \(\Gamma_g\) with holomorphic objects over \(M\). The interplay between these two methods is fascinating and suggestive.
A key role in the theory is that of a tight embedding of a symmetric space in \(\mathcal X\). A totally geodesic embedding is tight if it preserves maximality: the norm of the pullback of the Kähler form on \(\mathcal X\) is maximal. First appearing in Wienhard’s thesis, composition with tight embeddings is a basic construction for maximal representations. Holomorphic totally geodesic embeddings are tight, but not conversely.
Labourie defined the notion of an Anosov representation as one for which the geodesic flow on the unit tangent bundle \(U\Sigma_g\) admits a lift to the associated flat bundle of generalized flag manifolds \(G/B\) which preserves a hyperbolic splitting, where the Borel subgroup \(B\subset G\) is minimal parabolic. Maximal representations are Anosov in this sense, and this property implies many of the properties mentioned earlier when \(\mathcal X\) is the hyperbolic plane. In particular \(\rho\) is a quasi-isometric embedding and \(\rho(\gamma)\) is always hyperbolic. Furthermore \(\rho\) preserves a continuous (in fact Hölder) curve in the boundary \(G/P\). This curve is analogous to the hyperconvex curves in projective space \(\mathbb R\text{P}^n\) constructed by Labourie for Hitchin representations for \(G = \text{SL}(n + 1)\) and essentially coincides with it when \(G = \text{Sp}(4,\mathbb R) \subset \text{SL}(4,\mathbb R)\).
One of the main techniques of proof is the theory of continuous bounded continuous cohomology developed by Burger-Monod and Burger-Iozzi. The Milnor-Wood inequality was one of the origins of the theory of bounded cohomology. The Domic-Toledo inequality essentially expresses the fact that the Toledo invariant arises from a bounded continuous cohomology class on \(G\).
The paper concludes with a detailed study of the case when \(G = \text{Sp}(2n,\mathbb R)\). Starting from Fürstenberg’s theory of random walks and the Poisson formula, the \(\rho\)-equivariant measurable boundary map \(S^1\to G/P\) is discussed. (Here \(G/P\) is the Grassmannian of Lagrangian subspaces of \(\mathbb R^{2n}\).) Using the geometry of triples of Lagrangian subspaces, this map is shown to be continuous. From that the rigidity and quasi-isometry results follow.
The paper is clearly written with many examples. It provides a useful survey of a subject which interacts with many fields of mathematics, and provides a valuable exposition of this timely subject.

MSC:

53C35 Differential geometry of symmetric spaces
20F34 Fundamental groups and their automorphisms (group-theoretic aspects)
57M05 Fundamental group, presentations, free differential calculus
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