The framework of this paper is the so-called higher Teichmüller theory, that is, the extension of Teichmüller theory to representations of fundamental groups of surfaces to Lie groups that are more general than \(\mathrm{PSL}(2,\mathbb{R})\). The authors consider the case where \(G\) is a Lie group of Hermitian type.
Given a compact connected oriented surface \(\Sigma\), possibly with boundary, the authors define and study a bounded integer-valued function on the representation variety of \(\pi_1(\Sigma)\) into \(G\), which they call the Toledo invariant. The name was chosen because D. Toledo, in 1989, studied such a function in the particular setting of representations of fundamental groups of surfaces in complex hyperbolic spaces. The authors call maximal representations those whose Toledo invariant has maximal value. They develop the theory of maximal representations in this setting of a group \(G\) of Hermitian type.
The Toledo invariant is reminiscent of the Euler number, which is an integer-valued function defined on the representation variety \(\mathrm{Hom}(\pi_1(S), \mathrm{PSL}(2,\mathbb{R}))/\mathrm{PSL}(2,\mathbb{R})\) and which is constant on the connected components of the representation variety. The classical Teichmüller space is precisely the connected component with maximal Euler number (Goldman’s thesis, 1980).
In the setting of Lie groups of Hermitian type, the level set of the maximal value of the modulus of the Toledo invariant is a union of connected components of representations that the authors call maximal. The space of maximal representations is an instance of a higher Teichmüller space. Global properties of the components of the representation variety that have maximal Toledo invariant were studied from another point of view, in the case of closed surfaces, by Bradlow, García-Prada and Gothen, using Higgs bundles. In the paper under review, the authors establish properties of the Toledo invariant function on the representation variety such as continuity, uniform boundedness, additivity under connected sums of surfaces and congruence relations mod \(\mathbb{Z}\). They obtain information about the representation variety and geometric properties of maximal representations. They prove a structure theorem for such representations. They associate to maximal representations boundary maps with monotonicity (positivity) properties expressed in terms of maximal triples of points in the Shilov boundary of the symmetric space associated to the Lie group \(G\), and they give a formula for the Toledo invariant in terms of the bounded Euler class of a group action on a circle. The latter generalizes the notion of rotation number for circle homeomorphisms which was shown by Poincaré to be a semi-conjugacy invariant of oriention-preserving homeomorphisms of the circle.