Algebraic hull of maximal measurable cocycles of surface groups into Hermitian Lie groups
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Algebraic hull of maximal measurable cocycles of surface groups into Hermitian Lie groups A. Savini1 Received: 20 April 2020 / Accepted: 19 November 2020 © The Author(s) 2020
Abstract Following the work of Burger, Iozzi and Wienhard for representations, in this paper we introduce the notion of maximal measurable cocycles of a surface group. More precisely, let G be a semisimple algebraic R-group such that G = G(R)◦ is of Hermitian type. If ≤ L is a torsion-free lattice of a finite connected covering of PU(1, 1), given a standard Borel probability -space (, μ ), we introduce the notion of Toledo invariant for a measurable cocycle σ : × → G. The Toledo invariant remains unchanged along G-cohomology classes and its absolute value is bounded by the rank of G. This allows to define maximal measurable cocycles. We show that the algebraic hull H of a maximal cocycle σ is reductive and the centralizer of H = H(R)◦ is compact. If additionally σ admits a boundary map, then H is of tube type and σ is cohomologous to a cocycle stabilizing a unique maximal tube type subdomain. This result is analogous to the one obtained for representations. In the particular case G = PU(n, 1) maximality is sufficient to prove that σ is cohomologous to a cocycle preserving a complex geodesic. We conclude with some remarks about boundary maps of maximal Zariski dense cocycles. Keywords Hermitian Lie group · Tube type · Tightness · Shilov boundary · Maximal measurable cocycle · Kähler form · Toledo invariant
1 Introduction Given a torsion-free lattice ≤ L in a semisimple Lie group L, any representation ρ : → H into a locally compact group H induces a well-defined map at the level of continuous bounded cohomology groups. Hence fixed a preferred bounded class in the cohomology of H , one can pullback it and compare the resulting class with the fundamental class determined by via Kronecker pairing. This is a standard way to obtain numerical invariants for representations, whose importance has become evident in the study of rigidity and
The author was partially supported by the project Geometric and harmonic analysis with applications, funded by EU Horizon 2020 under the Marie Curie Grant Agreement No. 777822.
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A. Savini [email protected] Section de Mathématiques, University of Geneva, Rue Du Lièvre 2 1227, Switzerland
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Geometriae Dedicata
superrigidity properties. In many cases (such as the Toledo invariant, the Volume invariant or the Borel invariant) a numerical invariant has bounded absolute value and the maximum is attained if and only if the representation can be extended to a representation L → H of the ambient group. Several examples of these phenomena are given by the work of Bucher, Burger, Iozzi [2,3,29] in the case of representations of real hyperbolic lattices, by Burger and Iozzi [10] and by Duchesne and Pozzetti [16,40] for complex hyperbolic lattices and by the work of Burger, Iozzi and Wienhard [11–13] when the target group is of Hermitian type. In the latter case, of remarkable interest is the ana
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