A Matsumoto–mostow Result for Zimmer’s Cocycles of Hyperbolic Lattices

Abstract

Following the philosophy behind the theory of maximal representations, we introduce the volume of a Zimmer’s cocycle Γ × X → PO° (n, 1), where Γ is a torsion-free (non-)uniform lattice in PO° (n, 1), with n > 3, and X is a suitable standard Borel probability Γ-space. Our numerical invariant extends the volume of representations for (non-)uniform lattices to measurable cocycles and in the uniform setting it agrees with the generalized version of the Euler number of self-couplings. We prove that our volume of cocycles satisfies a Milnor–Wood type inequality in terms of the volume of the manifold Γ\ℍn. Additionally this invariant can be interpreted as a suitable multiplicative constant between bounded cohomology classes. This allows us to define a family of measurable cocycles with vanishing volume. The same interpretation enables us to characterize maximal cocycles for being cohomologous to the cocycle induced by the standard lattice embedding via a measurable map X → PO° (n, 1) with essentially constant sign.As a by-product of our rigidity result for the volume of cocycles, we give a different proof of the mapping degree theorem. This allows us to provide a complete characterization of maps homotopic to local isometries between closed hyperbolic manifolds in terms of maximal cocycles.In dimension n = 2, we introduce the notion of Euler number of measurable cocycles associated to a closed surface group and we show that it extends the classic Euler number of representations. Our Euler number agrees with the generalized version of the Euler number of self-couplings up to a multiplicative constant. Imitating the techniques developed in the case of the volume, we show a Milnor–Wood type inequality whose upper bound is given by the modulus of the Euler characteristic of the associated closed surface. This gives an alternative proof of the same result for the generalized version of the Euler number of self-couplings. Finally, using the interpretation of the Euler number as a multiplicative constant between bounded cohomology classes, we characterize maximal cocycles as those which are cohomologous to the one induced by a hyperbolization.

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

“…In this brief section we are going to recall the pullback determined by a boundary map associated to a measurable cocycle. We will actually introduce a more general way to define the pullback and we will show that it coincides with the approach introduced by the author and Moraschini [36,37] when a boundary map exists.…”

“…We want now to relate Definition 2.8 with the approach followed by the author and Moraschini in [36,37]. In the same setting of Definition 2.8, consider a minimal parabolic subgroup P ≤ G. Let (Y , ν) be any measure space such that the group H acts on Y by preserving the measure class of ν.…”

“…Recently the author [41] together with Moraschini [36,37] and Sarti [42] has applied bounded cohomology techniques to the study measurable cocycles with an essentially unique boundary map. The existence of a boundary map allows to define a pullback in bounded cohomology as in [7] and hence to develop a theory of numerical invariants, called multiplicative constants, also in the context of measurable cocycles.…”

“…Consider a standard Borel probability -space ( , μ ) and let G be a semisimple real algebraic group such that G = G(R) • is of Hermitian type. Using a measurable cocycle σ : × → G, we can define a pullback map in bounded cohomology and hence mimic the techniques used in [36,37] to define the Toledo invariant of σ . In the particular case that σ admits a boundary map φ : S 1 × →Š G into the Shilov boundary of G, then we recover the same approach developed in [36,37] (see Lemma 2.10).…”

“…Using a measurable cocycle σ : × → G, we can define a pullback map in bounded cohomology and hence mimic the techniques used in [36,37] to define the Toledo invariant of σ . In the particular case that σ admits a boundary map φ : S 1 × →Š G into the Shilov boundary of G, then we recover the same approach developed in [36,37] (see Lemma 2.10). In an analogous way to what happens for representations, the Toledo invariant is constant along G-cohomology classes and has absolute value bounded by rk(X ), the rank of the symmetric space X associated to G. Thus it makes sense to speak about maximal measurable cocycles.…”