Algebraic hull of maximal measurable cocycles of surface groups into Hermitian Lie groups

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 $$\mathbf {G}$$ G be a semisimple algebraic $${\mathbb {R}}$$ R -group such that $$G=\mathbf {G}({\mathbb {R}})^{\circ }$$ G = G ( R ) ∘ is of Hermitian type. If $$\Gamma \le L$$ Γ ≤ L is a torsion-free lattice of a finite connected covering of $$\mathrm{PU}(1,1)$$ PU ( 1 , 1 ) , given a standard Borel probability $$\Gamma $$ Γ -space $$(\Omega ,\mu _\Omega )$$ ( Ω , μ Ω ) , we introduce the notion of Toledo invariant for a measurable cocycle $$\sigma :\Gamma \times \Omega \rightarrow G$$ σ : Γ × Ω → 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 $$\mathbf {H}$$ H of a maximal cocycle $$\sigma $$ σ is reductive and the centralizer of $$H=\mathbf {H}({\mathbb {R}})^{\circ }$$ H = H ( R ) ∘ is compact. If additionally $$\sigma $$ σ admits a boundary map, then H is of tube type and $$\sigma $$ 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 $$G=\mathrm{PU}(n,1)$$ G = PU ( n , 1 ) maximality is sufficient to prove that $$\sigma $$ σ is cohomologous to a cocycle preserving a complex geodesic. We conclude with some remarks about boundary maps of maximal Zariski dense cocycles.

Superrigidity of maximal measurable cocycles of complex hyperbolic lattices

“…In this section we define the Toledo invariant associated to a measurable cocycle, generalizing the standard definition given for representations. Firstly, we do not use boundary maps, as done by the second author in [31] and then we follow the approach adopted by the second author and Moraschini in [26,28,30,31] for the definition of multiplicative constants. Indeed the Toledo invariant will be a particular case of multiplicative constant in the sense of [26].…”

“…Remark 4.1 Given a Zariski dense measurable cocycle σ : × X → G, where < PU(1, 1) is a lattice and G is a Hermitian Lie group, one the author [31,Section 4.1] conjectured that σ has a boundary map. Theorem 1 gives an answer to that question and has important consequences on the study of measurable cocycles of surface groups.…”

“…Theorem 1 gives an answer to that question and has important consequences on the study of measurable cocycles of surface groups. Indeed, in the case that σ is also maximal in the sense of Definition 3.5, then by [31,Theorem 1] the target must be a group of tube type. In this way one obtains a true generalization of [10,Theorem 5] to the context of measurable cocycles.…”

“…Finally notice that Theorem 1 has important consequences also on the work of one of the author [31], answering to the question of [31,Section 4.1] about the existence of boundary maps for Zariski dense measurable cocycles of surface groups.…”

“…Inspired by the work by Bader, Furman and Sauer [5] for couplings and by Burger and Iozzi [6] for representations, one of the author together with Moraschini [26,28,30,31] has recently developed the theory of numerical invariants of measurable cocycles, that is part of a program whose aim is to generalize rigidity results for representations of rank-one lattices. Those invariants are obtained by pulling back preferred classes in bounded cohomology along cocycles, imitating the case of representations.…”