---
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 $\mathbf{G}$ be a semisimple algebraic $\operatorname{\mathbb{R}}$-group such that $G=\mathbf{G}(\operatorname{\mathbb{R}})^\circ$ is of Hermitian type. If $\Gamma \leq L$ is a torsion-free lattice of a finite connected covering of $\operatorname{\textup{PU}}(1,1)$, given a standard Borel probability $\Gamma$-space $(\Omega,\mu_\Omega)$, we introduce the notion of Toledo invariant for a measurable cocycle $\sigma:\Gamma \times \Omega \rightarrow G$ with an essentially unique boundary map.

    The Toledo invariant is a multiplicative constant, hence it 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 $\mathbf{H}$ of a maximal cocycle $\sigma$ is reductive, the centralizer of $H=\mathbf{H}(\operatorname{\mathbb{R}})^\circ$ is compact, $H$ is of tube type and $\sigma$ is cohomologous to a cocycle stabilizing a unique maximal tube-type subdomain. This result is analogous to the one obtained for representations.

    We conclude with some remarks about boundary maps of maximal Zariski dense cocycles.
address: 'Department of Mathematics, University of Bologna, Piazza di Porta San Donato 5, 40126 Bologna, Italy'
author:
- 'A. Savini'
bibliography:
- 'biblionote.bib'
date: '. ©[ A. Savini 2020]{}'
title: Algebraic hull of maximal measurable cocycles of surface groups into Hermitian Lie groups
---

[^1]

.

Introduction
============

Given a torsion-free lattice $\Gamma \leq G$ in a semisimple Lie group $G$, any representation $\rho:\Gamma \rightarrow H$ 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 $\Gamma$ 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 superrigidity properties. Indeed a numerical invariant has bounded absolute value and the maximum is attained if and only if the representation can be extended to a representation $G \rightarrow H$ of the ambient group.

Several examples of these phenomena are given by the work of Bucher, Burger, Iozzi [@iozzi02:articolo; @bucher2:articolo; @BBIborel] in the case of representations of real hyperbolic lattices, by Burger and Iozzi [@BIcartan] and by Duchesne and Pozzetti [@Pozzetti; @duchesne:pozzetti] for complex hyperbolic lattices and by the work of Burger, Iozzi and Wienhard [@BIW07; @BIW09; @BIW1] when the target group is of Hermitian type. In the latter case, of remarkable interest is the analysis of the representation space $\textup{Hom}(\Gamma,G)$ when $G$ is a group of Hermitian type and $\Gamma$ is a lattice in a finite connected covering of $\operatorname{\textup{PU}}(1,1)$, that is a hyperbolic surface group. Burger, Iozzi and Wienhard [@BIW1] exploited the existence of a natural Kähler structure on the Hermitian symmetric space associated to $G$ in order to define the notion of *Toledo invariant* of a representation $\rho:\Gamma \rightarrow G$. That invariant has bounded absolute value and its maximality has important consequences on the Zariski closure $\mathbf{H}=\overline{\rho(\Gamma)}^Z$ of the image of the representation. Indeed the authors show that in the case of maximality $\mathbf{H}$ is reductive, $H=\mathbf{H}(\operatorname{\mathbb{R}})^\circ$ has compact centralizer and it is of tube type and the representation $\rho$ is injective with discrete image and it preserves a unique maximal tube-type subdomain [@BIW1 Theorem 5]. A domain is of *tube-type* if it can be written in the form $V+i\Omega$, where $V$ is a real vector space and $\Omega \subset V$ is an open convex cone. Maximal tube-type subdomains in a Hermitian symmetric space $\operatorname{\mathcal{X}}$ generalize the notion of complex geodesic in $\operatorname{\mathbb{H}}^n_{\operatorname{\mathbb{C}}}$ and they are all $G$-conjugated.

Partial results in the direction of [@BIW1 Theorem 5] were obtained by several authors. For instance when $G=\operatorname{\textup{PU}}(n,1)$ with $n \geq 2$, Toledo [@Toledo89] proved that maximal representations must preserve a complex geodesic. It is worth mentioning also the papers by Hernández [@Her91], by Koziarz and Maubon [@koziarz:maubon] and by Bradlow, García-Prada and Gothen [@garcia:geom; @garcia:dedicata]. In the latter case those results were obtained using different techniques based on the notion of Higgs bundle.

It is worth noticing that in the particular case of split real groups and surfaces without boundary, the set of maximal representations contains the Hitchin component [@hitchin]. The Hitchin component has been sistematically studied by serveral mathematicians. For instance Labourie [@labourie] focused his attention on the Asonov property, whereas Fock and Goncharov [@Fock:adv; @fock:hautes] related the Hitchin component with the notion of Lusztig’s positivity.

A crucial point in the proof of [@BIW1 Theorem 5] is that maximal representations are *tight*, that is the seminorm of the pullback class is equal to the norm of the bounded Kähler class. The tightness property has an analytic counterpart in terms of maps between symmetric spaces and Burger, Iozzi and Wienhard [@BIW09] give a complete characterization of tight subgroups of a Lie group of Hermitian type.

Recently the author [@savini3:articolo] together with Moraschini [@moraschini:savini; @moraschini:savini:2] and Sarti [@savini:sarti] 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 [@burger:articolo] and hence to develop a theory of numerical invariants, called *multiplicative constants*, also in the context of measurable cocycles.

The main goal of this paper is the study of measurable cocycles of surface groups. Let $\Gamma \leq L$ be a torsion-free lattice of a finite connected covering $L$ of $\operatorname{\textup{PU}}(1,1)$. Consider a standard Borel probability $\Gamma$-space $(\Omega,\mu_\Omega)$ and let $\mathbf{G}$ be a semisimple real algebraic group such that $G=\mathbf{G}(\operatorname{\mathbb{R}})^\circ$ is of Hermitian type. If a measurable cocycle $\sigma:\Gamma \times \Omega \rightarrow G$ admits an essentially unique boundary map $\phi:\operatorname{\mathbb{S}}^1 \times \Omega \rightarrow G$, then we can apply the theoretical background developed in [@moraschini:savini; @moraschini:savini:2] to defined the *Toledo invariant of $\sigma$*. In an analogous way to what happens for representations, the Toledo invariant is constant along $G$-cohomology classes and has absolute value bounded by $\operatorname{rk}(\operatorname{\mathcal{X}})$, the rank of the symmetric space $\operatorname{\mathcal{X}}$ associated to $G$. Thus it makes sense to speak about *maximal measurable cocycles*. This will be a particular example of *tight cocycles* (see Definition \[def:tight:cocycle\]).

Maximality allows to give a characterization of the *algebraic hull* of a measurable cocycle, as stated in the following

\[teor:maximal:alghull\] Let $\Gamma \leq L$ be a torsion-free lattice of a finite connected covering $L$ of $\operatorname{\textup{PU}}(1,1)$ and let $(\Omega,\mu_\Omega)$ be a standard Borel probability $\Gamma$-space. Let $\mathbf{G}$ be a semisimple algebraic $\operatorname{\mathbb{R}}$-group such that $G=\mathbf{G}(\operatorname{\mathbb{R}})^\circ$ is a Lie group of Hermitian type. Consider a measurable cocycle $\sigma:\Gamma \times \Omega \rightarrow G$ with essentially unique boundary map. Denote by $\mathbf{H}$ the algebraic hull of $\sigma$ in $\mathbf{G}$ and set $H=\mathbf{H}(\operatorname{\mathbb{R}})^\circ$. If $\sigma$ is maximal, then

1.  the algebraic hull $\mathbf{H}$ is reductive

2.  the centralizer $Z_G(H)$ is compact;

3.  the symmetric space $\operatorname{\mathcal{Y}}$ associated to $H$ is Hermitian of tube-type;

4.  it holds $\mathbf{H}(\operatorname{\mathbb{R}}) \subset \textup{Isom}(\operatorname{\mathcal{T}})$ for some maximal tube-type subdomain $\operatorname{\mathcal{T}}$ of $\operatorname{\mathcal{X}}$. Equivalently there exists a cocycle cohomologous to $\sigma$ which preserves $\operatorname{\mathcal{T}}$.

The above theorem should be interpreted as a suitable adaptation of [@BIW1 Theorem 5] to the context of maximal measurable cocycles. The first two properties are immediate consequences of the tightness of maximal cocycles, as shown in Theorem \[teor:alg:hull:tight\]. The tube-type condition is more involving and it is proved in Theorem \[teor:symmetric:tube\].

We conclude with some remarks about boundary maps of maximal Zariski dense cocycles. For representations, the relation between maximality and boundary maps preserving positivity of triples were studied by Guichard [@Guichard], Labourie [@labourie] and Fock and Goncharov [@fock:hautes]

Here we attempt to extend [@BIW1 Theorem 5.2] to the context of measurable cocycles. Given a maximal Zariski dense cocycle, we can construct a boundary map which has left-continuous (respectively right-continuous) slices. Moreover each slice preserves *transversality* and it is *monotone*, as proved in Theorem \[teor:boundary:map\]. Unfortunately, to get the statement, we need to make an additional assumption on the measurable map $\phi:\operatorname{\mathbb{S}}^1 \times \Omega \rightarrow \check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}}$. More precisely we need to assume that the essential image of almost every slice intersects nicely all closed algebraic subset of $\check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}}$ (Assumption \[ass:zariski:zero:measure\]). This assumption is clearly verified by cocycles cohomologous to maximal Zariski dense representations, but we do not know more generally under which conditions of both $\sigma$ or $\phi$ this is true and it would be interesting to know it. The proof of Theorem \[teor:boundary:map\] follows the line of [@BIWL Section 8] and of [@BIW1 Theorem 5.2].

Plan of the paper {#plan-of-the-paper .unnumbered}
-----------------

In Section \[sec:preliminary\] we recall the preliminary definitions and results that we need in the paper. In Section \[sec:measurable:cocycles\] we remind the notion of measurable cocycle and of cohomology class determined by a cocycle. Of particular importance for our purpose will be the definition of algebraic hull. Then we conclude the section with some elements of boundary theory. Section \[sec:burger:monod\] is devoted to continuous and continuous bounded cohomology. We remind the functorial approach by Burger and Monod and we recall the definition of pullback induced by a boundary map. The last part is devoted to Hermitian symmetric spaces (Section \[sec:hermitian:groups\]).

The main theorem of paper is proved in Section \[sec:maximal:cocycles\]. We first introduce the notion of Toledo invariant of a measurable cocycle in Section \[sec:toledo:invariant\]. In Section \[sec:maximal:cocycle:thm\] it appears the definition of maximal cocycle. Maximal cocycles are tight by Proposition \[prop:maximal:tight\] and this result togheter with Theorem \[teor:symmetric:tube\] allows to prove Theorem \[teor:maximal:alghull\]. We conclude with Section \[sec:boundary:map\], where we prove Theorem \[teor:boundary:map\].

Preliminary definitions and results {#sec:preliminary}
===================================

Measurable cocycles {#sec:measurable:cocycles}
-------------------

The following section is devoted to a quick review about measurable cocycles theory. We are going to recall the definitions of both measurable cocycle and cohomology class. Then we will introduce the notion of algebraic hull and we will conclude the section with some elements of boundary theory. For a more detailed discussion about those topics we refer the reader to the work of both Furstenberg [@furst:articolo73; @furst:articolo] and Zimmer [@zimmer:preprint; @zimmer:annals; @zimmer:libro].

Consider two locally compact second countable groups $G,H$ endowed with their Haar measurable structure. Given a standard Borel measure space $(\Omega,\mu_\Omega)$ we say that it is a *$G$-space* if $G$ acts on $\Omega$ by measure-preserving transformations. Additionally if $\mu_\Omega$ is a probability measure, we are going to call $(\Omega,\mu_\Omega)$ a *standard Borel probability $G$-space*. Given another measure space $(\Theta,\nu)$, we are going to denote by $\textup{Meas}(X,Y)$ the space of measurable functions with the topology of the convergence in measure.

\[def:measurable:cocycle\] Let $G,H$ two locally compact second countable groups and let $(\Omega,\mu_\Omega)$ be a standard Borel probability $G$-space. A measurable function $\sigma:G \times \Omega \rightarrow H$ is a *measurable cocycle* if it holds $$\label{eq:measurable:cocycle}
\sigma(g_1 g_2,s)=\sigma(g_1,g_2 s)\sigma(g_2,s) \ ,$$ for almost every $g_1,g_2 \in G$ and almost every $s \in \Omega$.

Measurable cocycles are quite ubiquitous in mathematics and Equation can be suitably interpreted as a naive generalization to the measurable context of the chain rule for differentiation of smooth functions. By writing a measurable cocycle $\sigma$ as an element $\sigma \in \textup{Meas}(G,\textup{Meas}(\Omega,H))$, Equation boils down the cocycle condition. Indeed $\sigma$ may be interpreted as a Borel $1$-cocycle in the sense of Eilenberg-MacLane (see [@feldman:moore; @zimmer:preprint] for more details about this interpretation). Following this line, one could naturally ask when two different cocycles are cohomologous.

\[def:cohomologous:cocycles\] Let $\sigma:G \times \Omega \rightarrow H$ be a measurable cocycle and let $f:\Omega \rightarrow H$ be a measurable function. The *$f$-twisted cocycle of $\sigma$* is defined as $$\sigma^f:G \times \Omega \rightarrow H, \ \ \sigma^f(g,s):=f(gs)^{-1}\sigma(g,s)f(s) \ .$$ We say that two cocycles $\sigma_1,\sigma_2:G \times \Omega \rightarrow H$ are *cohomologous* if there exists a measurable function $f:\Omega \rightarrow H$ such that $$\sigma_2^f=\sigma_1 \ .$$

Choosing a measurable function $f:\Omega \rightarrow H$ is a typical way to construct cocycles starting from representations. Indeed, given a continuous representation $\rho:G \rightarrow H$, one can verifiy that the measurable function $$\sigma_\rho:G \times \Omega \rightarrow H \ , \ \ \sigma_\rho(g,s):=\rho(g) \ ,$$ is a measurable cocycle as a consequence of the morphism condition. This allows to see representation theory into the wider world of measurable cocycles theory. Additionally this offers us the possibility to interpret the notion of cohomologous cocycles as a generalization of conjugated representations.

Given a representation $\rho:G \rightarrow H$, if the image is not closed, it is quite natural to consider its closure, which it is still a subgroup of $H$. Unfortunately the image of a cocycle has no structure a priori. Nevertheless, if $H$ corresponds to the real points of a real algebraic group, then there is a notion which is in some sense similar to take the closure of the image of a representation.

Suppose that $\mathbf{H}$ is a real algebraic group. Let $\sigma:G \times \Omega \rightarrow \mathbf{H}(\operatorname{\mathbb{R}})$ be a measurable cocycle. The *algebraic hull associated to $\sigma$* is (the conjugacy class of) the smallest algebraic subgroup $\mathbf{L}$ of $\mathbf{H}$ such that $\mathbf{L}(\operatorname{\mathbb{R}})$ contains the image of a cocycle cohomologous to $\sigma$.

As proved in [@zimmer:libro Proposition 9.2] this notion is well-defined by the descending chain condition on algebraic subgroups and it depends only the cohomology class of the cocycle.

We conclude this brief discussion about measurable cocycle introducing some elements of boundary theory. In order to do this, we are going to assume that $G$ is a semisimple Lie group of non-compact type. Let $Q$ be any parabolic subgroup of $G$ and let $Y$ be a measurable $H$-space.

\[def:boundary:map\] Let $\sigma:G \times \Omega \rightarrow H$ be a measurable cocycle. A *(generalized) boundary map* is a measurable map $\phi:G/Q \times \Omega \rightarrow Y$ which is $\sigma$-equivariant, that is $$\phi(g \xi,g s)=\sigma(g,s)\phi(\xi,s) \ ,$$ for every $g \in G$ and almost every $\xi \in G/Q, s \in \Omega$.

It is easy to check that, if $\phi:G/Q \times \Omega \rightarrow Y$ is a boundary map for $\sigma$, then $\phi^f:G/Q \times \Omega \rightarrow Y, \ \phi^f(\xi,s):=f(s)^{-1}\phi(\xi,s)$ is a boundary map for $\sigma^f$ for any measurable function $f:\Omega \rightarrow H$.

The existence and the uniqueness of a boundary map associated to a cocycle $\sigma$ rely on the dynamical properties of $\sigma$. For a more detailed discussion about it we refer the reader to [@furst:articolo]. Boundary maps for measurable cocycles will be crucial to define a pullback map in bounded cohomology imitating the work done by Burger and Iozzi [@burger:articolo] in the case of representations.

Continuous bounded cohomology and functorial approach {#sec:burger:monod}
-----------------------------------------------------

Given a locally compact group $G$ we are going to remind the notion of continuous and continuous bounded cohomology groups of $G$. A remarkable aspect of continuous bounded cohomology is that it can be computed using any strong resolution by relatively injective modules. We are going to give few details about this functorial apprach and we will conclude the section by introducing the notion of pullback along a boundary map associated to a measurable cocycle. For more details about continuous bounded cohomology and its functorial approach we refer to the work of Burger and Monod [@burger2:articolo; @monod:libro], whereas we refer to the papers of the author together with Moraschini [@moraschini:savini; @moraschini:savini:2] for more details about pullback along boundary maps.

Consider a *Banach $G$-module* $E$, that is $E$ is a Banach space with an isometric action $\pi:G \rightarrow \textup{Isom}(E)$. We are going to assume that $E$ is the dual of some separable Banach space, so that it makes sense to speak about the weak-${}^\ast$ topology on $E$.

We consider the set $$\begin{aligned}
\operatorname{\textup{C}}^\bullet_{cb}(G;E):=\{ f : G^{\bullet+1} \rightarrow E \ | & \ f \ \textup{is continuous and} \\
&\lVert f \rVert_\infty:=\sup_{g_0,\ldots,g_{\bullet}}\lVert f(g_0,\ldots,g_{\bullet}) \rVert_E < \infty \} \ , \end{aligned}$$ where $\lVert \ \cdot \ \rVert_E$ denotes the norm on the space $E$. Each $\operatorname{\textup{C}}^\bullet_{cb}(G;E)$ is a normed via the supremum norm and it can be endowed with an isometric action by $G$ defined by $$(gf)(g_0,\ldots,g_\bullet):=\pi(g)f(g^{-1}g_0,\ldots,g^{-1}g_\bullet) \ ,$$ where $f \in \operatorname{\textup{C}}^\bullet_{cb}(G;\operatorname{\mathbb{R}})$ and $g,g_0,\ldots,g_\bullet \in G$. Defining the *standard homogeneous coboundary operator* by $$\delta^\bullet:\operatorname{\textup{C}}^\bullet_{cb}(G;E) \rightarrow \operatorname{\textup{C}}^{\bullet+1}_{cb}(G;E) \ ,$$ $$\delta^\bullet(f)(g_0,\ldots,g_{\bullet+1}):=\sum_{i=0}^{\bullet+1}(-1)^i f(g_0,\ldots,\hat g_i,\ldots,g_{\bullet+1}) \ ,$$ we get a cochain complex $(\operatorname{\textup{C}}^\bullet_{cb}(G;E),\delta^\bullet)$.

\[def:bounded:cohomology\] Let $G$ be a locally compact group and let $E$ be a Banach $G$-module. The *$k$-th continuous bounded cohomology group* of $G$ with coefficients in $E$ is the $k$-th cohomology group of the $G$-invariant subcomplex $(\operatorname{\textup{C}}^\bullet_{cb}(G;E)^G,\delta^\bullet)$, that is $$\operatorname{\textup{H}}^k_{cb}(G;E):=\operatorname{\textup{H}}^k(\operatorname{\textup{C}}^\bullet_{cb}(G;E)^G) \ ,$$ for every $k \geq 0$.

It is worth noticing that each cohomology group $\operatorname{\textup{H}}^\bullet_{cb}(G;E)$ has a natural seminormed structure inherited by the normed structure on the continuous bounded cochains.

By dropping the assumption of boundedness one can define similarly the complex of continuous cochains $(\operatorname{\textup{C}}^\bullet_c(G;E),\delta^\bullet)$ and the standard inclusion $i:\operatorname{\textup{C}}^\bullet_{cb}(G;E) \rightarrow \operatorname{\textup{C}}^\bullet_c(G;E)$ induces a map at a cohomological level $$\textup{comp}^\bullet:\operatorname{\textup{H}}^\bullet_{cb}(G;E) \rightarrow \operatorname{\textup{H}}^\bullet_c(G;E) \ ,$$ called *comparison map*.

Computing continuous bounded cohomology of a locally compact group $G$ using only the definition given above may reveal quite difficult. For this reason Burger and Monod [@burger2:articolo; @monod:libro] introduced a way to compute continuous bounded cohomology groups based on the notion of resolutions. More precisely the authors showed [@burger2:articolo Corollary 1.5.3] that given any Banach $G$-module $E$ and any strong resolution $(E^\bullet,d^\bullet)$ of $E$ by relatively injective Banach $G$-modules, it holds $$\operatorname{\textup{H}}^k_{cb}(G;E) \cong \operatorname{\textup{H}}^k((E^\bullet)^G) \ ,$$ for every $k \geq 0$. Since we will not need the notion of strong resolution and of relatively injective Banach $G$-module, we omit them and we refer to the book of Monod [@monod:libro] for more details.

Unfortunately the isomorphism given above it is not isometric a priori, that is it may not preserve the seminormed structure. Nevertheless there are specific resolutions for which the isomorphism it is actually isometric. This is the case for instance when we consider the resolution of essentially bounded weak-$^\ast$ measurable functions $(\operatorname{\textup{L}}^\infty_{\textup{w}^\ast}((G/Q)^{\bullet+1};E),\delta^\bullet)$ on the quotient $G/Q$ [@burger2:articolo Theorem 1], where $G$ is a semisimple Lie group of non-compact type and $Q$ is any parabolic subgroup. We are going to exploit this result for the Shilov boundary of a Hermitian symmetric space.

We conclude this brief section by recalling the pullback determined by a boundary map associated to a measurable cocycle. Suppose that $G$ is a semisimple Lie group of non-compact type and consider a parabolic subgroup $Q \leq G$. Let $(\Omega,\mu_\Omega)$ be a standard Borel probability $G$-space and let $Y$ be any measurable $H$-space, where $H$ is another locally compact group. Denote by $(\operatorname{\mathcal{B}}^\infty_{\textup{w}^\ast}(Y^{\bullet+1};E),\delta^\bullet)$ the complex of weak-$^\ast$ measurable bounded functions on $Y$ (with the injection of coefficients, the latter is a strong resolution of $E$ by [@burger:articolo Proposition 2.1]). Given a boundary map $\phi:G/Q \times \Omega \rightarrow Y$ associated to a measurable cocycle $\sigma:G \times \Omega \rightarrow H$, there exists a natural map defined at the level of cochains as $$\operatorname{\textup{C}}^\bullet(\Phi^\Omega):\operatorname{\mathcal{B}}^\infty(Y^{\bullet+1};E)^H \rightarrow \operatorname{\textup{L}}^\infty((G/Q)^{\bullet+1};E)^G \ ,$$ $$\operatorname{\textup{C}}^\bullet(\Phi^\Omega)(\psi)(\xi_0,\ldots,\xi_\bullet):=\int_{\Omega} \psi(\phi(\xi_0,s),\ldots,\phi(\xi_\bullet,s))d\mu_\Omega(s) \ .$$

As shown by the author and Moraschini [@moraschini:savini; @moraschini:savini:2], the above map is a chain map which does not increase the norm and it induces a well define map in cohomology $$\operatorname{\textup{H}}^\bullet(\Phi^\Omega):\operatorname{\textup{H}}^\bullet(\operatorname{\mathcal{B}}(Y^{\bullet+1};E)^H) \rightarrow \operatorname{\textup{H}}^\bullet_{cb}(G;E) \ , \ \operatorname{\textup{H}}^\bullet(\Phi^\Omega)([\psi]):=[\operatorname{\textup{C}}^\bullet(\Phi^\Omega)(\psi)] \ .$$ The map $\operatorname{\textup{H}}^\bullet(\Phi^\Omega)$ is the *pullback induced by the boundary map $\phi$*. We are going to use the pullback map in order to define properly the Toledo invariant of a measurable cocycle of a surface group.

Lie groups of Hermitian type {#sec:hermitian:groups}
----------------------------

In this section we are going to recall the main definitions and result about Lie groups of Hermitian type. We are going to remind the notion of Shilov boundary for a Hermitian symmetric space and we are going to define a suitable cocycle on it, called Bergmann cocycle, which will enable us to define the notion of maximality for measurable cocycles of surface groups. For a more detailed discussion about these notions, we refer mainly to the work of Burger, Iozzi and Wienhard [@BIW07; @BIW09; @BIW1].

\[def:hermitian:symmetric:space\] Let $\operatorname{\mathcal{X}}$ be a Riemannian symmetric space and denote by $G=\textup{Isom}(\operatorname{\mathcal{X}})^\circ$ the connected component of the identity of the isometry group associated to $\operatorname{\mathcal{X}}$. We say that $\operatorname{\mathcal{X}}$ is *Hermitian* if there exists a $G$-invariant complex structure $\operatorname{\mathcal{J}}$ on $\operatorname{\mathcal{X}}$. Given a semisimple real algebraic group $\mathbf{G}$, we say that $G=\mathbf{G}(\operatorname{\mathbb{R}})^\circ$ is *of Hermitian type* if its symmetric space $\operatorname{\mathcal{X}}$ is Hermitian.

Among all the possible ones, a family of examples of particular interest in this paper will be the one of Hermitian symmetric spaces of tube-type. We say that a Hermitian symmetric space $\operatorname{\mathcal{X}}$ is *of tube-type* if it is biholomorphic to a complex subset of the form $V+i\Omega$, where $V$ is a real vector space and $\Omega \subset V$ is a proper convex cone. A typical example is given by the hyperbolic space $\operatorname{\mathbb{H}}^2$ associated to the group $\operatorname{\textup{PU}}(1,1)$, or more generally to the symmetric space associated to $\operatorname{\textup{PU}}(p,p)$ when $p \geq 2$.

A Hermitian symmetric space $\operatorname{\mathcal{X}}$ can be bihomolorphically realized as bounded convex domain $\operatorname{\mathcal{D}}_{\operatorname{\mathcal{X}}}$ in $\operatorname{\mathbb{C}}^n$. For such a realization, the group $G=\textup{Isom}(\operatorname{\mathcal{X}})^\circ$ acts via biholomorphisms and its action can be extended in a continuous way to the boundary $\partial \operatorname{\mathcal{D}}_{\operatorname{\mathcal{X}}}$. Unfortunately the latter is not a homogeneous $G$-space, but it admits a unique closed $G$-orbit. The latter will be identified with the Shilov boundary.

More precisely we give first the following

\[def:shilov:boundary\] Let $\operatorname{\mathcal{D}}\subset \operatorname{\mathbb{C}}^n$ be a bounded domain. The *Shilov boundary* $\check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{D}}}$ of $\operatorname{\mathcal{D}}$ is the unique closed subset of $\partial \operatorname{\mathcal{D}}$ such that, given a function $f$ continuous on $\overline{\operatorname{\mathcal{D}}}$ and holomorphic on $\operatorname{\mathcal{D}}$, then $$\max_{\overline{D}}|f|=\max_{\check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{D}}}} |f| \ .$$ Given a Hermitian symmetric space $\operatorname{\mathcal{X}}$, we denote by $\check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}}$ the Shilov boundary associated to the bounded realization of $\operatorname{\mathcal{X}}$ and we call it *the Shilov boundary of $\operatorname{\mathcal{X}}$*.

As already anticipated the Shilov boundary associated to a Hermitian symmetric space $\operatorname{\mathcal{X}}$ is a homogeneous $G$-space. Indeed if we denote by $\mathbf{G}$ the algebraic group associated to the complexified Lie algebra of $G=\textup{Isom}(\operatorname{\mathcal{X}})^\circ$, then there exists a maximal parabolic subgroup $\mathbf{Q} \subset \mathbf{G}$ such that $\check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}}$ can be identified with $\mathbf{G}/\mathbf{Q}(\operatorname{\mathbb{R}})$. In particular $\check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}}$ is an amenable $G$-space in the sense of Section \[sec:burger:monod\] and hence the resolution of essentially bounded functions on it computes isometrically the continuous bounded cohomology of $G$.

In order to describe more accurately the second bounded cohomology group of $G$, recall that if $\operatorname{\mathcal{X}}$ is Hermitian, then there exists a $G$-invariant complex structure $\operatorname{\mathcal{J}}$ on it. If we denote by $g$ the $G$-invariant Riemannian metric on $\operatorname{\mathcal{X}}$, we can define the *Kähler form* $$(\omega_{\operatorname{\mathcal{X}}})_a(X,Y):=g_a(X,\operatorname{\mathcal{J}}_a Y) \ ,$$ for any $X,Y \in T_a \operatorname{\mathcal{X}}$. Being $G$-invariant, the form $\omega_{\operatorname{\mathcal{X}}}$ is automatically closed by Cartan’s Lemma. Define now $$\label{eq:cocycle:symmetric:space}
\beta_{\operatorname{\mathcal{X}}}: (\operatorname{\mathcal{X}})^{(3)} \rightarrow \operatorname{\mathbb{R}}, \ \ \beta_{\operatorname{\mathcal{X}}}(a_1,a_2,a_3):=\int_{\Delta(a_1,a_2,a_3)} \omega_{\operatorname{\mathcal{X}}} \ ,$$ where $\Delta(a_1,a_2,a_3)$ is any triangle with geodesic sides determined by $a_1,a_2,a_3 \in \operatorname{\mathcal{X}}$. Since $\omega_{\operatorname{\mathcal{X}}}$ is closed, by Stokes Theorem the function $\beta_{\operatorname{\mathcal{X}}}$ is an alternating $G$-invariant cocycle on $\operatorname{\mathcal{X}}$. Remarkably the cocycle $\beta_{\operatorname{\mathcal{X}}}$ can be extended to a strict measurable $G$-invariant cocycle on the Shilov boundary $\check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}}$ [@BIW07 Corollary 3.8] and its absolute value is bounded $\operatorname{rk}(\operatorname{\mathcal{X}})$. We are going to denote such an extension with $\beta_{\operatorname{\mathcal{X}}}$ with an abuse of notation. As previously said in Section \[sec:burger:monod\] the cocycle $\beta_{\operatorname{\mathcal{X}}} \in \operatorname{\textup{L}}^\infty((\check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}})^{(3)};\operatorname{\mathbb{R}})^G$ determines canonically a class in $\operatorname{\textup{H}}^2_{cb}(G;\operatorname{\mathbb{R}})$.

We call *Bergmann cocycle* the measurable extension $\beta_{\operatorname{\mathcal{X}}}: \check{\operatorname{\mathcal{S}}}^{(3)} \rightarrow \operatorname{\mathbb{R}}$ to the Shilov boundary of the cocycle defined by Equation .

We denote by $\kappa^b_G \in \operatorname{\textup{H}}^2_{cb}(G;\operatorname{\mathbb{R}})$ the class determined by $\beta_{\operatorname{\mathcal{X}}}$ and we call it *bounded Kähler class*.

Recall that two points $\xi,\eta \in \check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}}$ are *transverse* if they lie in the unique open $G$-orbit in $(\check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}})^2$. We conclude the section by recalling some properties of the Bergmann cocycle when $\operatorname{\mathcal{X}}$ is a Hermitian symmetric space of tube-type. As stated in [@BIW1 Lemma 5.5], if $\operatorname{\mathcal{X}}$ is of tube-type then

1.  the cocycle $\beta_{\operatorname{\mathcal{X}}}$ takes values in the discrete set $\{-\frac{\operatorname{rk}(\operatorname{\mathcal{X}})}{2}, - \frac{\operatorname{rk}(\operatorname{\mathcal{X}})}{2}+1 , \ldots , \frac{\operatorname{rk}(\operatorname{\mathcal{X}})}{2}-1, \frac{\operatorname{rk}(\operatorname{\mathcal{X}})}{2} \}$;

2.  if it holds $|\beta_{\operatorname{\mathcal{X}}}(\xi,\eta,\omega)|=\frac{\operatorname{rk}(\operatorname{\mathcal{X}})}{2}$, then $\xi,\eta,\omega$ are pairwise transverse;

3.  we can decompose $$(\check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}})^{(3)}= \sqcup_{i=0}^{\operatorname{rk}(\operatorname{\mathcal{X}})} \operatorname{\mathcal{O}}_{-\operatorname{rk}(\operatorname{\mathcal{X}})+2i} \ ,$$ where $\operatorname{\mathcal{O}}_{-\operatorname{rk}(\operatorname{\mathcal{X}})+2i}$ is the open subset of $(\check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}})^{(3)}$ where $\beta_{\operatorname{\mathcal{X}}}$ is identically equal to $-\frac{\operatorname{rk}(\operatorname{\mathcal{X}})}{2}+i$;

4.  given $\xi, (\xi_n)_{n \in \operatorname{\mathbb{N}}}, (\xi'_n)_{n \in \operatorname{\mathbb{N}}}$ where $\xi,\xi_n,\xi'_n \in \check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}}$, if $\lim_{n \to \infty} \xi_n =\xi$ and $\beta_{\operatorname{\mathcal{X}}}(\xi,\xi_n,\xi'_n)=\frac{\operatorname{rk}(\operatorname{\mathcal{X}})}{2}$ then $\lim_{n \to \infty} \xi'_n=\xi$.

Maximal measurable cocycles of surface groups {#sec:maximal:cocycles}
=============================================

Let $L$ be a finite connected covering of the group $\operatorname{\textup{PU}}(1,1)$ and consider a torsion-free lattice $\Gamma \leq L$. Let $(\Omega,\mu_\Omega)$ be a standard Borel probability $\Gamma$-space. Given an irreducible Hermitian symmetric space $\operatorname{\mathcal{X}}$, we are going to denote by $G=\text{Isom}^\circ(\operatorname{\mathcal{X}})$ the connected component of the identity of the isometry group of $\operatorname{\mathcal{X}}$. In this section we are going to introduce the notion of *Toledo invariant* associated to a measurable cocycle $\sigma:\Gamma \times \Omega \rightarrow G$ with an essentially unique boundary map. Since this numerical invariant will have absolute value bounded from above by the rank of $\operatorname{\mathcal{X}}$, it will make sense to talk about *maximal cocycles*. Maximal cocycles will be particular examples of *tight cocycles*. We are going to introduce the notion of *tightness* which will have important consequences on the algebraic hull.

We are going to show that if a maximal cocycle is Zariski dense then the Hermitian symmetric space $\operatorname{\mathcal{X}}$ must be of tube-type. Hence there are no maximal Zariski dense cocycle in Hermitian Lie group which are not of tube-type. Moreover, maximality affects also the regularity property of the slices of boundary maps.

The Toledo invariant of a measurable cocycle {#sec:toledo:invariant}
--------------------------------------------

Let $L$ be a finite connected covering of the group $\operatorname{\textup{PU}}(1,1)$ and consider a torsion-free lattice $\Gamma \leq L$. Let $(\Omega,\mu_\Omega)$ be a standard Borel probability $\Gamma$-space. Denote by $G=\text{Isom}^\circ(\operatorname{\mathcal{X}})$ the connected component of the identity of the isometry group of an irreducible Hermitian symmetric space $\operatorname{\mathcal{X}}$. Let $\mathbf{G}$ be the connected Lie group associated to the complexified Lie algebra of $G$, so that $G=\mathbf{G}(\operatorname{\mathbb{R}})^\circ$. Let $\sigma:\Gamma \times \Omega \rightarrow G$ be a measurable cocycle with essentially unique boundary map $\phi:\operatorname{\mathbb{S}}^1 \times \Omega \rightarrow \check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}}$. Here $\check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}}$ is the Shilov boundary associated to the symmetric space $\operatorname{\mathcal{X}}$.

Recall by Section \[sec:burger:monod\] that the existence of the boundary map $\phi$ induces a pullback map in cohomology $$\operatorname{\textup{H}}^\bullet(\Phi^\Omega): \operatorname{\textup{H}}^\bullet(\operatorname{\mathcal{B}}^\infty((\check{S}_{\operatorname{\mathcal{X}}})^{\bullet+1};\operatorname{\mathbb{R}})^G) \rightarrow \operatorname{\textup{H}}^\bullet_b(\Gamma;\operatorname{\mathbb{R}}) \ .$$ In particular we are allowed to consider the pullback of the Bergmann cocycle $\beta_{\operatorname{\mathcal{X}}}$. Since $\Gamma$ is a lattice of $L$, we have a well-defined *transfer map*, which is given at the level of cochains by $$\hat{\operatorname{\textup{T}}}^\bullet_b:\operatorname{\textup{L}}^\infty((\operatorname{\mathbb{S}}^1)^{\bullet+1};\operatorname{\mathbb{R}})^\Gamma \rightarrow \operatorname{\textup{L}}^\infty((\operatorname{\mathbb{S}}^1)^{\bullet+1};\operatorname{\mathbb{R}})^L \ ,$$ $$\hat{\operatorname{\textup{T}}}^\bullet_b(\psi)(\xi_0,\ldots,\xi_\bullet):=\int_{\Gamma \backslash L} \psi(\overline{g}\xi_0,\ldots,\overline{g}\xi_\bullet)d\mu_{\Gamma \backslash L}(\overline{g}) \ ,$$ where $\overline{g}$ denotes the equivalence class of $g$ in $\Gamma \backslash L$ and $\mu_{\Gamma \backslash L}$ is the normalized $L$-invariant measure on the quotient. Being a chain map, $\hat \operatorname{\textup{T}}^\bullet_b$ induces a well-defined map in cohomology called *transfer map* $$\operatorname{\textup{T}}^\bullet_b:\operatorname{\textup{H}}^\bullet_b(\Gamma;\operatorname{\mathbb{R}}) \rightarrow \operatorname{\textup{H}}^\bullet_{cb}(L;\operatorname{\mathbb{R}}), \hspace{5pt} \operatorname{\textup{T}}^\bullet_b([\psi]):=[\hat \operatorname{\textup{T}}^\bullet_b(\psi)] \ .$$

It is worth recalling that the bounded Kähler class $\kappa^b_L$ is a generator of the group $\operatorname{\textup{H}}^2_{cb}(L;\operatorname{\mathbb{R}})$ and it is represented by the Bergmann cocycle on the circle $\beta_{\operatorname{\mathbb{S}}^1}$ (which is nothing else than the orientation cocycle). In this particular setting, we are allowed to give the following

\[def:toledo:inv\] Let $\Gamma \leq L$ be a torsion-free lattice and $(\Omega,\mu_\Omega)$ a standard Borel probability $\Gamma$-space. Consider a measurable cocycle $\sigma:\Gamma \times \Omega \rightarrow G$ with essentially unique boundary map $\phi:\operatorname{\mathbb{S}}^1 \times \Omega \rightarrow \check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}}$. The *Toledo invariant* $\textup{t}_b(\sigma)$ associated to $\sigma$ is defined as $$\label{eq:toledo}
\operatorname{\textup{T}}^2_b \circ \operatorname{\textup{H}}^2(\Phi^\Omega)([ \beta_{\operatorname{\mathcal{X}}}])=\textup{t}_b(\sigma)[\beta_{\operatorname{\mathbb{S}}^1}]=\textup{t}_b(\sigma)\kappa^b_L \ .$$

Since the $\Gamma$-action on the circle is doubly ergodic and the cocycles are alternating, Equation holds actually at the level of cochains, that is $$\begin{aligned}
\label{eq:formula}
\int_{\Gamma \backslash L} \int_\Omega & \beta_{\operatorname{\mathcal{X}}}(\phi(\overline{g}\xi,s),\phi(\overline{g}\eta,s),\phi(\overline{g}\omega,s))d\mu_\Omega(s)d\mu_{\Gamma \backslash L}(\overline{g}) =\\
=& \text{t}_b(\sigma)\beta_{\operatorname{\mathbb{S}}^1}(\xi,\eta,\omega) \nonumber \  ,\end{aligned}$$ and the equation holds for *every* $\xi,\eta,\omega \in \operatorname{\mathbb{S}}^1$, as a consequence of either Burger and Iozzi [@BIcartan] or Pozzetti [@Pozzetti], for instance. Notice that Equation is simply a suitable adaptation of [@BIW1 Corollary 4.4] to the context of measurable cocycles.

It is immediate to verify that the Toledo invariant is a *multiplicative constant* in the sense of [@moraschini:savini:2 Definition 3.16]. Indeed following the notation of that paper, the setting required by [@moraschini:savini:2 Definition 3.16] is satisfied and one has $$\textup{t}_b(\sigma)=\lambda_{\beta_{\operatorname{\mathcal{X}}},\beta_{\operatorname{\mathbb{S}}^1}}(\sigma) \ .$$ Thanks to this analogy, one can immediately argue that $\textup{t}_b(\sigma)$ is invariant along the $G$-cohomology class of $\sigma$ and its absolute value can be bounded from above as follows $$|\text{t}_b(\sigma)| \leq \operatorname{rk}(\operatorname{\mathcal{X}}) \ .$$

\[oss:alternative:definition\] We could have defined the Toledo invariant in a different way. Let $\Gamma \leq L$ be a torsion-free lattice and let $(\Omega,\mu_\Omega)$ be a standard Borel probability $\Gamma$-space. Denote by $\Sigma$ the finite-area surface obtained as the quotient of $\operatorname{\mathbb{H}}^2_{\operatorname{\mathbb{R}}}$ by $\Gamma$, that is $\Sigma=\Gamma \backslash \operatorname{\mathbb{H}}^2_{\operatorname{\mathbb{R}}}$. If $\Gamma$ is *uniform* we know that $\Sigma$ is closed, whereas when $\Gamma$ is *non-uniform* then the surface $\Sigma$ has finitely many cusps. In the latter case we are going to denote by $S$ a *compact core* of $\Sigma$, otherwise we set $S=\Sigma$.

Following [@moraschini:savini Section 3.4] we can define the following composition of functions $$\label{eq:j:composition}
\operatorname{\textup{J}}^\bullet_{S, \partial S}: \operatorname{\textup{H}}^\bullet_b(\Gamma;\operatorname{\mathbb{R}}) \rightarrow \operatorname{\textup{H}}^\bullet_b(\Sigma;\operatorname{\mathbb{R}}) \rightarrow \operatorname{\textup{H}}^\bullet_b(\Sigma,\Sigma \setminus S) \rightarrow \operatorname{\textup{H}}^\bullet_b(S,\partial S) \ ,$$ where the first map is the isomorphism given by the Gromov’s Mapping Theorem [@Grom82; @Ivanov; @FM:grom], the second map is obtained by the long exact sequence in bounded cohomology [@BBFIPP] and the last map is induced by the homotopy equivalence $(\Sigma,\Sigma \setminus S) \simeq (S, \partial S)$.

Given a measurable cocycle $\sigma:\Gamma \times \Omega \rightarrow G$ with essentially unique boundary map $\phi:\operatorname{\mathbb{S}}^1 \times \Omega \rightarrow \check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}}$, we could have defined the *Toledo number* of the cocycle $\sigma$ as $$\operatorname{\textup{T}}_b(\sigma):= \langle \textup{comp}^2_{S, \partial S} \circ \operatorname{\textup{J}}^2_{S, \partial S} \circ \operatorname{\textup{H}}^2(\Phi^\Omega)([\beta_{\operatorname{\mathcal{X}}}]),[S,\partial S] \rangle$$

To compare the two different definitions of the Toledo invariant, one can follows the same strategy of the proofs of either [@moraschini:savini Proposition 1.2, Proposition 1.6] or [@moraschini:savini:2 Proposition 5.5]. In this way it is possible to show that $$\label{eq:alternative:toledo}
\textup{t}_b(\sigma)=\frac{\textup{T}_b(\sigma)}{|\chi(\Sigma)|} \ ,$$ where $\chi(\Sigma)$ is the Euler characteristic of the surface $\Sigma$. Notice that Equation is analogous to the one obtained by Burger, Iozzi and Wienhard [@BIW1 Theorem 3.3]. In particular $\textup{T}_b(\sigma)$ is an invariant of the $G$-cohomology class of $\sigma$ and it holds the following estimate $$|\textup{T}_b(\sigma)| \leq \operatorname{rk}(\operatorname{\mathcal{X}}) |\chi(\Sigma)| \ .$$

Maximal measurable cocycles of surface groups {#sec:maximal:cocycle:thm}
---------------------------------------------

In this section we are going to introduce the notion of maximality. Maximal measurable cocycles represent the first example of tight cocycles and this has important consequences on their algebraic hull. Additionally, if they are Zariski dense then the target must be a Hermitian Lie group of tube-type.

We start by giving the definition of maximality.

\[def:maximal:cocycle\] Let $\Gamma \leq L$ be a torsion-free lattice and let $(\Omega,\mu_\Omega)$ be a standard Borel probability $\Gamma$-space. Consider a measurable cocycle $\sigma:\Gamma \times \Omega \rightarrow G$ with essentially unique boundary map. We say that $\sigma$ is *maximal* if it holds $\text{t}_b(\sigma)=\operatorname{rk}(\operatorname{\mathcal{X}})$.

In order to show that maximal cocycles are tight, we need first to introduce the notion of tightness for measurable cocycles of surface groups. Inspired by the notion for representations studied by Burger, Iozzi and Wienhard [@BIW09], we can give the following

\[def:tight:cocycle\] Let $\Gamma \leq L$ be a torsion-free lattice and $(\Omega,\mu_\Omega)$ a standard Borel probability $\Gamma$-space. Consider a measurable cocycle $\sigma:\Gamma \times \Omega \rightarrow G$ with essentially unique boundary map $\phi:\operatorname{\mathbb{S}}^1 \times \Omega \rightarrow \check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}}$. We say that $\sigma$ is *tight* if it holds $$\lVert \operatorname{\textup{H}}^2(\Phi^\Omega)([\beta_{\operatorname{\mathcal{X}}}]) \rVert_\infty=\frac{\operatorname{rk}\operatorname{\mathcal{X}}}{2} \ .$$

Clearly the definition above mimic the one given for representations. Indeed it is immediate to check that if the cocycle is cohomologous to the one induced by a representation, Definition \[def:tight:cocycle\] boils down to the standard one. Another important aspect is that the tightness property is invariant along the $G$-cohomology class of a given cocycle [@moraschini:savini:2 Proposition 3.12]. Notice that we could have introduced the notion of tightness in a much more general setting, but this would be not so useful for our purposes.

The deep study of tight representations done by Burger, Iozzi and Wienhard [@BIW09] enables us to state the following theorem which characterizes the algebraic hull of a tight cocycle and which is a direct consequence of [@BIW09 Theorem 3], where a full characterization of tight subgroups is given.

\[teor:alg:hull:tight\] Let $\Gamma$ be a torsion-free lattice of a finite connected covering $L$ of $\operatorname{\textup{PU}}(1,1)$ and let $(\Omega,\mu_\Omega)$ be a standard Borel probability $\Gamma$-space. Consider $\mathbf{G}$ a semisimple algebraic $\operatorname{\mathbb{R}}$-group such that $G=\mathbf{G}(\operatorname{\mathbb{R}})^\circ$ is a Lie group of Hermitian type. Given a measurable cocycle $\sigma:\Gamma \times \Omega \rightarrow G$, assume that there exists an essentially unique boundary map $\phi:\operatorname{\mathbb{S}}^1 \times \Omega \rightarrow \check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}}$. Denote by $\mathbf{H}$ the algebraic hull of $\sigma$ in $\mathbf{G}$ and set $H=\mathbf{H}(\operatorname{\mathbb{R}})^\circ$. If $\sigma$ is tight then

1.  $\mathbf{H}$ is a reductive group;

2.  the centralizer $Z_G(H)$ is compact;

3.  if $\operatorname{\mathcal{Y}}$ is the symmetric space associated to $H$, there exists a unique $H$-invariant complex structure on $\operatorname{\mathcal{Y}}$ such that the inclusion $H \rightarrow G$ is tight and positive.

Since the cocycle is tight and this condition is invariant along the $G$-cohomology class of $\sigma$, the inclusion $i:H \rightarrow G$ is tight. The conclusion follows by direct application of [@BIW09 Theorem 7.1] which characterize tight subgroups of $G$.

The next step is to prove that maximal cocycles are tight in the sense of Definition \[def:tight:cocycle\], similarly for what happens in the case of representations [@BIW1 Lemma 6.1]. This result will have important consequence for the algebraic hull of a maximal cocycle as a direct application of Theorem \[teor:alg:hull:tight\].

\[prop:maximal:tight\] Let $\Gamma \leq L$ be a torsion-free lattice and let $(\Omega,\mu_\Omega)$ be a standard Borel probability $\Gamma$-space. Consider a measurable cocycle $\sigma:\Gamma \times \Omega \rightarrow G$ with essentially unique boundary map. If $\sigma$ is maximal then it is tight.

Suppose that $\sigma:\Gamma \times \Omega \rightarrow G$ is maximal. Then it holds $\textup{t}_b(\sigma)=\operatorname{rk}\operatorname{\mathcal{X}}$. By definition we have that $$\textup{T}^2_b \circ \textup{H}^2_b(\Phi^\Omega)([\beta_{\operatorname{\mathcal{X}}}])=\operatorname{rk}(\operatorname{\mathcal{X}}) \kappa^b_L \ ,$$ and hence it follows $$\operatorname{rk}(\operatorname{\mathcal{X}}) = \lVert \operatorname{rk}(\operatorname{\mathcal{X}}) \kappa^b_L \rVert_\infty=\lVert \textup{T}^2_b \circ \textup{H}^2_b(\Phi^\Omega)([\beta_{\operatorname{\mathcal{X}}}]) \rVert \leq \rVert \textup{H}^2_b(\Phi^\Omega)([\beta_{\operatorname{\mathcal{X}}}]) \rVert_\infty \ .$$ Since the pullback is norm non-increasing, we have also that $\lVert \textup{H}^2_b(\Phi^\Omega)([\beta_{\operatorname{\mathcal{X}}}]) \rVert_\infty \leq \operatorname{rk}(\operatorname{\mathcal{X}})$, whence we must have equality and the cocycle $\sigma$ is tight.

Before moving in our discussion, we need to remind briefly some notation regarding the triple products studied by Burger, Iozzi and Wienhard [@BIW07]. If we denote by $(\check{S}_{\operatorname{\mathcal{X}}})^{(3)}$ the set of triples of distinct points in $\check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}}$, the *Hermitian triple product* is defined as

$$\langle \langle \cdot , \cdot , \cdot \rangle \rangle: (\check{S}_ {\operatorname{\mathcal{X}}})^{(3)} \rightarrow \operatorname{\mathbb{R}}^\times \backslash \operatorname{\mathbb{C}}^\times \ ,$$ $$\langle\langle \xi,\eta,\omega \rangle\rangle=e^{i \pi p_{\operatorname{\mathcal{X}}} \beta_{\operatorname{\mathcal{X}}}(\xi,\eta,\omega)} \ \  \text{mod} \operatorname{\mathbb{R}}^\times \ ,$$ for every $(\xi,\eta,\omega) \in \check{S}^{(3)}_{\operatorname{\mathcal{X}}}$. The number $p_{\operatorname{\mathcal{X}}}$ is an integer defined in terms of the root system associated to $G$.

Recall that $\check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}}$ is a homogeneous $G$-space, which can be realized as the quotient $G/Q$, where $Q=\mathbf{Q}(\operatorname{\mathbb{R}})$ and $\mathbf{Q}$ is a maximal parabolic subgroup of $\mathbf{G}$. Burger, Iozzi and Wienhard were able to extend the Hermitian triple product to a *complex Hermitian triple product* $\langle\langle \cdot, \cdot, \cdot \rangle\rangle_{\operatorname{\mathbb{C}}}$ defined on $(\mathbf{G}/\mathbf{Q})^3$ with values into $\Delta^\times \backslash A^\times$. Here $A^\times$ is the group $\operatorname{\mathbb{C}}^\times \times \operatorname{\mathbb{C}}^\times$ endowed with real structure $(\lambda,\mu) \mapsto (\overline{\mu},\overline{\lambda})$ and $\Delta^\times$ is the image of the diagonal embedding of $\operatorname{\mathbb{C}}^\times$. More precisely, the authors [@BIW07 Section 2.4] showed that the following diagram commutes $$\xymatrix{
(\check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}})^{(3)} \ar[rr]^{\langle \langle \cdot, \cdot, \cdot \rangle\rangle} \ar[d]^{(\imath)^3} && \operatorname{\mathbb{R}}^\times \backslash \operatorname{\mathbb{C}}^\times \ar[d]^\Delta\\
(\mathbf{G}/\mathbf{Q})^3 \ar[rr]^{\langle \langle \cdot, \cdot, \cdot \rangle\rangle_{\operatorname{\mathbb{C}}}} && \Delta^\times \backslash A^\times \ .
}$$ where $\imath:\check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}} \rightarrow \mathbf{G}/\mathbf{Q}$ is the map given by the $G$-equivariant identification of $\check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}}$ with $(\mathbf{G}/\mathbf{Q})(\operatorname{\mathbb{R}})$ and $\Delta$ is the diagonal embedding.

Given any pair of distinct points $(\xi,\eta) \in (\check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}})^{(2)}$, following [@BIW07 Section 5.1], we denote by $\operatorname{\mathcal{O}}_{\xi,\eta}$ the open Zariski subset of $\mathbf{G}/\mathbf{Q}$ on which the map $$p_{\xi,\eta}:\operatorname{\mathcal{O}}_{\xi,\eta} \rightarrow \Delta^\times \backslash A^\times, \hspace{5pt} p_{\xi,\eta}(\omega):=\langle \langle \xi, \eta, \omega \rangle \rangle_{\operatorname{\mathbb{C}}} \ ,$$ is well-defined. Burger, Iozzi and Wienhard [@BIW07 Lemma 5.1] proved that if there exists an integer $m \in \operatorname{\mathbb{Z}}\setminus \{ 0 \}$ such that $\omega \mapsto p_{\xi,\eta}(\omega)^m$ is constant, then $\operatorname{\mathcal{X}}$ must be of tube-type.

Now we can proceed proving the following theorem, which should be thought of as a generalization of [@BIW1 Theorem 4.1(1)].

\[teor:symmetric:tube\] Let $L$ be a finite connected covering of $\operatorname{\textup{PU}}(1,1)$ and let $\Gamma \leq L$ be a torsion-free lattice. Let $(\Omega,\mu_\Omega)$ be a standard Borel probability $\Gamma$-space and let $\sigma:\Gamma \times \Omega \rightarrow G$ be a measurable cocycle with essentially unique boundary map. If $\sigma$ is maximal and Zariski dense, then $\operatorname{\mathcal{X}}$ must be of tube-type.

Consider a positively oriented triple of distinct points $\xi,\eta,\omega \in \operatorname{\mathbb{S}}^1$. By the maximality assumption we have that $\textup{t}_b(\sigma)=\operatorname{rk}(\operatorname{\mathcal{X}})$ and by substituting this value in Equation we obtain $$\label{eq:formula:maximal}
\int_{\Gamma \backslash L}\int_\Omega \beta_{\operatorname{\mathcal{X}}}(\phi(\overline{g}\xi,s),\phi(\overline{g}\eta,s),\phi(\overline{g}\omega,s))d\mu_\Omega(s)d\mu_{\Gamma \backslash L}(\overline{g})=\frac{\operatorname{rk}(\operatorname{\mathcal{X}})}{2} \ .$$ Hence for almost every $\overline{g} \in \Gamma \backslash L$ and almost every $s \in \Omega$ it holds $$\beta_{\operatorname{\mathcal{X}}}(\phi(\overline{g}\xi,s),\phi(\overline{g}\eta,s),\phi(\overline{g}\omega,s))=\frac{\operatorname{rk}(\operatorname{\mathcal{X}})}{2} \ ,$$ and by the equivariance of the map $\phi$ it follows $$\label{eq:almost:every:maximal}
\beta_{\operatorname{\mathcal{X}}}(\phi(g\xi,s),\phi(g\eta,s),\phi(g\omega,s))=\frac{\operatorname{rk}(\operatorname{\mathcal{X}})}{2} \ ,$$ for almost every $g \in L$ and almost every $s \in \Omega$.

For almost every $s \in \Omega$, we know that the $s$-slice $\phi_s:\operatorname{\mathbb{S}}^1 \rightarrow \check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}}$ is measurable and, by Equation \[eq:almost:every:maximal\] it satisfies $$\label{eq:maximal:slice}
\beta_{\operatorname{\mathcal{X}}}(\phi_s(g\xi),\phi_s(g\eta),\phi_s(g\omega))=\frac{\operatorname{rk}(\operatorname{\mathcal{X}})}{2} \ ,$$ for almost every $g \in L$. Since the same reasoning applies to a negatively oriented triple, we must have $$\label{eq:maximal:slice:triples}
\beta_{\operatorname{\mathcal{X}}}(\phi_s(\xi),\phi_s(\eta),\phi_s(\omega))=\pm \frac{\operatorname{rk}(\operatorname{\mathcal{X}})}{2} \ ,$$ for almost every triple $\xi,\eta,\omega$ such that $\beta_{\operatorname{\mathbb{S}}^1}(\xi,\eta,\omega)=\pm 1/2$. The equation above implies that $$\label{eq:hermitian:product}
\langle \langle \phi_s(\xi),\phi_s(\eta),\phi_s(\omega) \rangle\rangle^2 = 1 \ \  \textup{mod} \operatorname{\mathbb{R}}^\times \ ,$$ for almost every $\xi,\eta,\omega \in \operatorname{\mathbb{S}}^1$ distinct.

Fix now a pair $(\xi,\eta) \in (\operatorname{\mathbb{S}}^1)^{2}$ such that Equation holds for almost every $\omega \in \operatorname{\mathbb{S}}^1$. Since $G$ acts transitively on the set of maximal triples on $\check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}}$ by [@BIW09 Theorem 3.8(3)], for a fixed pair of distinct points $\xi_0,\eta_0 \in \check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}}$, there exists a measurable function $f:\Omega \rightarrow G$ such that $$\phi_s(\xi)=f(s)\xi_0 \ , \hspace{10pt} \phi_s(\eta)=f(s)\eta_0 \ , \ (\xi_0,\eta_0,f(s)^{-1}\phi_s(\omega)) \ \text{is maximal}$$ for almost every $\omega \in \operatorname{\mathbb{S}}^1, s \in \Omega$. For such a measurable function $f$, we consider $\sigma^f$ and the map $\phi^f$ as the ones defined in Section \[sec:measurable:cocycles\]. For the ease of notation we are going to write $\alpha=\sigma^f$ and $\psi=\phi^f$. By the choice of the map $f$, Equation can be rewritten as $$\langle \langle \xi_0,\eta_0,\psi_s(\omega) \rangle\rangle^2 = 1 \ \  \textup{mod} \operatorname{\mathbb{R}}^\times \ ,$$ for almost every $\omega \in \operatorname{\mathbb{S}}^1$. The previous equation implies that $\psi_s(\omega) \in \operatorname{\mathcal{O}}_{\xi_0,\eta_0}$ for almost every $\omega \in \operatorname{\mathbb{S}}^1$ and almost every $s \in \Omega$. We denote by $E$ the subset of full measure in $\operatorname{\mathbb{S}}^1 \times \Omega$ such that $\psi_s(\omega) \in \operatorname{\mathcal{O}}_{\xi_0,\eta_0}$ for all $E$. Define $$E^\Gamma:=\bigcap_{\gamma \in \Gamma} \gamma E \ ,$$ which has full measure being a countable intersection of full measure sets (notice that $\Gamma$ preserves the measure class on $\operatorname{\mathbb{S}}^1 \times \Omega$). Since $\sigma$ is Zariski dense, the cocycle $\alpha$ is Zariski dense too. Since the Zariski closure of $\psi(E^\Gamma)$ is preserved by the algebraic hull of $\alpha$ which coincides with $\mathbf{G}$, the set $\psi(E^\Gamma)$ is Zariski dense in $\mathbf{G}/\mathbf{Q}$, whence is $\psi(E^\Gamma)$ Zariski dense in $\operatorname{\mathcal{O}}_{\xi_0,\eta_0}$. Thus the map $\omega \rightarrow p_{\xi_0,\eta_0}(\omega)^2$ is constant on $\operatorname{\mathcal{O}}_{\xi_0,\eta_0}$ and $\operatorname{\mathcal{X}}$ is of tube-type, as claimed.

An important consequence of the previous theorem is the following

Let $L$ be a finite connected covering of $\operatorname{\textup{PU}}(1,1)$ and let $\Gamma \leq L$ be a torsion-free lattice. Let $(\Omega,\mu_\Omega)$ be a standard Borel probability $\Gamma$-space. There is no maximal Zariski dense cocycle $\sigma:\Gamma \times \Omega \rightarrow G$ when $G$ is not of tube-type.

As a consequence of Theorem \[teor:symmetric:tube\], if $\mathbf{H}$ is the algebraic hull of a maximal cocycle $\sigma$ and $H=\mathbf{H}(\operatorname{\mathbb{R}})$, then $H^\circ$ must be a Hermitian group of tube-type.

The following theorem collects all the properties we discovered about the algebraic hull of a maximal cocycle and it should be thought of as a statement equivalent to [@BIW1 Theorem 5] in the context of measurable cocycles.

[teor:maximal:alghull]{} Let $\Gamma \leq L$ be a torsion-free lattice and let $(\Omega,\mu_\Omega)$ be a standard Borel probability $\Gamma$-space. Let $\mathbf{G}$ be a semisimple algebraic $\operatorname{\mathbb{R}}$-group such that $G=\mathbf{G}(\operatorname{\mathbb{R}})^\circ$ is a Lie group of Hermitian type. Consider a measurable cocycle $\sigma:\Gamma \times \Omega \rightarrow G$ with essentially unique boundary map. Denote by $\mathbf{H}$ the algebraic hull of $\sigma$ in $\mathbf{G}$ and set $H=\mathbf{H}(\operatorname{\mathbb{R}})^\circ$. If $\sigma$ is maximal, then

1.  the algebraic hull $\mathbf{H}$ is reductive

2.  the centralizer $Z_G(H)$ is compact;

3.  the symmetric space $\operatorname{\mathcal{Y}}$ associated to $H$ is Hermitian of tube-type;

4.  it holds $\mathbf{H}(\operatorname{\mathbb{R}}) \subset \textup{Isom}(\operatorname{\mathcal{T}})$ for some maximal tube-type subdomain $\operatorname{\mathcal{T}}$ of $\operatorname{\mathcal{X}}$. Equivalently there exists a cocycle cohomologous to $\sigma$ which preserves $\operatorname{\mathcal{T}}$.

Being maximal, the cocycle $\sigma$ is tight by Proposition \[prop:maximal:tight\]. Hence we can apply Theorem \[teor:alg:hull:tight\] to get properties $1)$ and $2)$. Additionally by Theorem \[teor:symmetric:tube\] the symmetric space $\operatorname{\mathcal{Y}}$ must be of tube-type, whence point $3)$.

The inclusion $i: H \rightarrow G$ is tight because the cocycle $\sigma$ is tight. Since the symmetric space $\operatorname{\mathcal{Y}}$ associated to $H$ is of tube-type and the inclusion is tight, by [@BIW09 Theorem 9(1)] there exists a unique maximal tube-type subdomain $\operatorname{\mathcal{T}}$ of $\operatorname{\mathcal{X}}$ preserved by $H$. By uniqueness, $\operatorname{\mathcal{T}}$ must be preserved by the whole $\mathbf{H}(\operatorname{\mathbb{R}})$ and we are done.

Regularity properties of boundary maps {#sec:boundary:map}
--------------------------------------

Imitating what happens in the context of representations, we are going to study the regularity properties of boundaries map associated to maximal measurable cocycles. Given a maximal Zariski dense measurable cocycle, under suitable hypothesis on the push-forward measure with respect to the slices of the boundary map, we are going to show that there exists an essentially unique equivariant measurable map with left-continuous (respectively right-continuous) slices which preserve transversality and maximality. We are going to follow the line of [@BIW1 Section 5].

Before introducing the setup of the section, we say that a measurable map $\phi:\operatorname{\mathbb{S}}^1 \rightarrow \check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}}$ is *maximal* if it satisfies Equation . Notice that almost every slice of a boundary map associated to a maximal cocycle is maximal.

\[setup:boundary:map\] From now until the end of the section we are going to assume the following

-   $\Gamma \leq L$ be a torsion-free lattice of a finite connected covering $L$ of $\operatorname{\textup{PU}}(1,1)$;

-   $(\Omega,\mu_\Omega)$ is a standard Borel probability $\Gamma$-space;

-   $\sigma:\Gamma \times \Omega \rightarrow G$ is a maximal Zariski dense cocycle with boundary map $\phi:\operatorname{\mathbb{S}}^1 \times \Omega \rightarrow \check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}}$;

-   denote by $\{E_s\}_{s \in \Omega}$ the family of essential graphs $E_s=\textup{EssGr}(\phi_s)$ associated to the slices, that is the support of the push-forward of the Lebesgue measure on $\operatorname{\mathbb{S}}^1$ with respect to the map $\xi \mapsto  (\xi,\phi_s(\xi)) \in \operatorname{\mathbb{S}}^1 \times \check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}}$.

Having introduced the setup we needed, we can now move on proving the following

\[lemma:maximality:triples\] In the situation of Setup \[setup:boundary:map\], suppose that $E_s$ is maximal. Let $(\xi_i,\eta_i) \in E_s$ for $i=1,2,3$ be points such that $\xi_1,\xi_2,\xi_3$ are pairwise distinct and $\eta_1,\eta_2,\eta_3$ are pairwise transverse. Then it holds $$\beta_{\operatorname{\mathcal{X}}}(\eta_1,\eta_2,\eta_3)=\operatorname{rk}(\operatorname{\mathcal{X}}) \beta_{\operatorname{\mathbb{S}}^1}(\xi_1,\xi_2,\xi_3) \ .$$

Denote by $I_i$ for $i=1,2,3$ open paiwise non-intersecting intervals such that $\xi_i \in I_i$ and for any $\omega_i \in I_i$ it holds $$\beta_{\operatorname{\mathbb{S}}^1}(\omega_1,\omega_2,\omega_3)=\beta_{\operatorname{\mathbb{S}}^1}(\xi_1,\xi_2,\xi_3) \ .$$ Consider a open neighborhood $U_i$ of $\eta_i$, for $i=1,2,3$, such that $U_1 \times U_2 \times U_3 \in (\check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}})^{(3)}$. Then the measurable set $$A_i=\{ \omega \in I_i \ | \ \phi_s(\omega_i) \in U_i \} \ ,$$ is a set of positive measure, since $\eta_1,\eta_2,\eta_3$ are in the essential image of $\phi_s$. Since we assumed the slice $E_s$ is maximal, for almost every $(\omega_1,\omega_2,\omega_3) \in A_1 \times A_2 \times A_3$ we have that $$\beta_{\operatorname{\mathcal{X}}}(\phi_s(\omega_1),\phi_s(\omega_2),\phi_s(\omega_3))=\operatorname{rk}(\operatorname{\mathcal{X}})\beta_{\operatorname{\mathbb{S}}^1}(\omega_1,\omega_2,\omega_3)=\operatorname{rk}(\operatorname{\mathcal{X}})\beta_{\operatorname{\mathbb{S}}^1}(\xi_1,\xi_2,\xi_3) \ .$$

By setting $\varepsilon=2\beta_{\operatorname{\mathbb{S}}^1}(\xi_1,\xi_2,\xi_3)$, we have that $|\varepsilon|=1$ and for almost every $(\omega_1,\omega_2,\omega_3) \in A_1 \times A_2 \times A_3$ we have that $$(\phi_s(\omega_1),\phi_s(\omega_2),\phi_s(\omega_3)) \in U_1 \times U_2 \times U_3 \cap \operatorname{\mathcal{O}}_{\varepsilon \operatorname{rk}{\operatorname{\mathcal{X}}}} \ ,$$ where $\operatorname{\mathcal{O}}_{\varepsilon \operatorname{rk}{\operatorname{\mathcal{X}}}}$ is the open set in $\check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}}^3$ on which $\beta_{\operatorname{\mathcal{X}}}$ is identically equal to $\operatorname{rk}(\operatorname{\mathcal{X}})/2$. By the arbitrary choice of the neighborhood $U_i$, must have $(\eta_1,\eta_2,\eta_3) \in \overline{\operatorname{\mathcal{O}}_{\varepsilon \operatorname{rk}\operatorname{\mathcal{X}}}}$.

Since we have that $$\overline{\operatorname{\mathcal{O}}_{\varepsilon \operatorname{rk}\operatorname{\mathcal{X}}}} \cap (\check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}})^{(3)}=\overline{\operatorname{\mathcal{O}}_{\varepsilon \operatorname{rk}\operatorname{\mathcal{X}}}} \cap ( \sqcup_{i=0}^{\operatorname{rk}(\operatorname{\mathcal{X}})}  \operatorname{\mathcal{O}}_{-\operatorname{rk}\operatorname{\mathcal{X}}+ 2 i}) = \operatorname{\mathcal{O}}_{\varepsilon \operatorname{rk}\operatorname{\mathcal{X}}}$$ and $(\eta_1,\eta_2,\eta_3) \in (\check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}})^{(3)}$, the triple is maximal and the claim follows.

In order to proceed we have now to discuss a condition we have to impose on the slices of the boundary map. Recall that $\check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}}$ can be identified with $G/Q$, where $Q$ is a maximal parabolic subgroup. We denote by $\textbf{V}_\xi \subset \mathbf{G}/\mathbf{Q}$ the Zariski closed set of points transverse to $\xi$ and set $V_\xi:=\textbf{V}_\xi(\operatorname{\mathbb{R}})$, the set of points transvers to $\xi$ in the Shilov boundary.

Burger, Iozzi and Wienhard [@BIW1 Proposition 5.2] proved that the boundary map associated to a Zariski dense representation has very strong properties, since its essential image intersects any proper Zariski closed set of the Shilov boundary in a set of measure zero. The author wonder under which hypothesis the same property should hold for almost every slice of a boundary map associated to a cocycle. Here we are going to assume it. More precisely

\[ass:zariski:zero:measure\] In the situation of Setup \[setup:boundary:map\], we suppose that for every propery Zariski closed set $\mathbf{V} \subset \mathbf{G}/\mathbf{Q}$ it holds $$\nu(\phi_s^{-1}(\mathbf{V}(\operatorname{\mathbb{R}})))=0 \ ,$$ for almost every $s \in \Omega$. Here $\nu$ is the round measure on $\operatorname{\mathbb{S}}^1$.

Assumption \[ass:zariski:zero:measure\] is clearly satisfied by cocycles which are cohomologous to a Zariski dense representation $\rho:\Gamma \rightarrow G$. We are not aware if this property can be extended to a wider class of cocycles.

\[lemma:transverse\] Let $E_s$ be a maximal slice satisfying Assumption \[ass:zariski:zero:measure\] and let $(\xi_1,\eta_1), (\xi_2,\eta_2) \in E_s$ with $\xi_1 \neq \xi_2$. Then $\eta_1$ and $\eta_2$ are transverse.

For any distinct $\xi,\omega \in \operatorname{\mathbb{S}}^1$ we denote by $$((\xi,\omega)):=\{ \eta \in \operatorname{\mathbb{S}}^1 \ | \ \beta_{\operatorname{\mathbb{S}}^1}(\xi,\zeta,\omega)=\frac{1}{2} \} \ .$$ Thanks to Assumption \[ass:zariski:zero:measure\], we can suppose that the essential image of the slice $\phi_s$ meets any Zariski closed set in a measure zero set. Hence we can find $\alpha_1 \in ((\xi_1,\xi_2))$ such that $\phi_s(\alpha_1)$ is transverse to both $\eta_1$ and $\eta_2$. In the same way there will exist a point $\alpha_2 \in ((\xi_2,\xi_1))$ such that $\phi_s(\alpha_2)$ is transverse to $\eta_1$ and $\eta_2$.

Using now jointly Lemma \[lemma:maximality:triples\] and the cocycle condition on $\beta_{\operatorname{\mathcal{X}}}$ we get $$\begin{aligned}
0&=\beta_{\operatorname{\mathcal{X}}}(\phi_s(\alpha_1),\eta_1,\phi_s(\alpha_2))-\beta_{\operatorname{\mathcal{X}}}(\eta_1,\eta_2,\phi_s(\alpha_2))+\\
 &+\beta_{\operatorname{\mathcal{X}}}(\eta_1,\phi_s(\alpha_1),\phi_s(\alpha_2))-\beta_{\operatorname{\mathcal{X}}}(\eta_1,\phi_s(\alpha_1),\eta_2))=\\
 &=\frac{\operatorname{rk}(\operatorname{\mathcal{X}})}{2}-\beta_{\operatorname{\mathcal{X}}}(\eta_1,\eta_2,\phi_s(\alpha_2))+\frac{\operatorname{rk}(\operatorname{\mathcal{X}})}{2}-\beta_{\operatorname{\mathcal{X}}}(\eta_1,\phi_s(\alpha_1),\eta_2)) \ . \\\end{aligned}$$ The previous line implies that $\beta_{\operatorname{\mathcal{X}}}(\eta_1,\eta_2,\phi_s(\alpha_2))=\frac{\operatorname{rk}(\operatorname{\mathcal{X}})}{2}$ and hence $\eta_1$ and $\eta_2$ are transverse.

Given now any subset $A \subset \operatorname{\mathbb{S}}^1$ we put $$F_A^s:=\{ \eta \in \check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}} \  | \ \exists \  \xi \in A \ : \ (\xi,\eta) \in E_s \} \ .$$ We define also $$((\xi,\omega]]:=((\xi,\omega)) \cup \{ \omega \} \ .$$

\[lemma:one:point\] Let $s \in \Omega$ be a point such that $E_s$ is a maximal slice satisfying Assumption \[ass:zariski:zero:measure\]. Let $\xi \neq \omega$ be two points in $\operatorname{\mathbb{S}}^1$. Then $\overline{F^s_{((\xi,\omega]]}} \cap F^s_\xi$ and $\overline{F^s_{[[\omega,\xi))}} \cap F^s_{\xi}$ consist each of one point.

We prove that $\overline{F^s_{((\xi,\omega]]}} \cap F^s_\xi$ consists of exactly one point. The same strategy can be applied to $\overline{F^s_{[[\omega,\xi))}} \cap F^s_{\xi}$ to prove the same statement.

Let $\eta,\eta' \in \overline{F^s_{((\xi,\omega]]}} \cap F^s_\xi$ and consider $(\xi_n,\eta_n) \in E_s$ a sequence such that $$\xi_n \in ((\xi,\omega]], \ \ \ \lim_{n \to \infty} \xi_n=\xi, \ \ \ \lim_{n \to \infty} \eta_n=\eta \ .$$

Given any $\zeta \in ((\xi,\omega))$, we can apply the same reasoning of [@BIW1 Lemma 5.8], to say that $$\overline{F^s_{((\xi,\omega]]}} \cap F^s_\xi = \overline{F^s_{((\xi,\zeta]]}} \cap F^s_\xi \ .$$ Thanks to the previous equation, consider a sequence $(\omega_n,\eta'_n) \in E_s$ so that $$\omega_n \in ((\xi,\xi_n)), \ \ \  \lim_{n \to \infty} \omega_n=\xi, \ \ \ \lim_{n \to \infty} \eta'_n=\eta' \ .$$

Applying Lemma \[lemma:transverse\] we have that $\eta,\eta'_n,\eta_n$ are pairwise transverse. Hence we can apply Lemma \[lemma:maximality:triples\] to the triples $(\xi,\omega_n,\xi_n)$ and $(\eta,\eta_n',\eta_n)$ to get $$\beta_{\operatorname{\mathcal{X}}}(\eta,\eta_n',\eta_n)=\operatorname{rk}(\operatorname{\mathcal{X}})\beta_{\operatorname{\mathbb{S}}^1}(\xi,\omega_n,\xi_n)=\frac{\operatorname{rk}(\operatorname{\mathcal{X}})}{2} \ .$$ Since $\lim_{n \to \infty} \eta_n=\eta$, Property $4)$ of Section \[sec:hermitian:groups\] of the Bergmann cocycles $\beta_{\operatorname{\mathcal{X}}}$ forces $\lim_{n \to \infty} \eta'_n=\eta$ and hence $\eta=\eta'$.

In this way wet get immediately the following

\[cor:two:points\] Let $s \in \Omega$ be a point such that $E_s$ is a maximal slice satisfying Assumption \[ass:zariski:zero:measure\]. For every $\xi \in \operatorname{\mathbb{S}}^1$ the set $F^s_\xi$ contains either one or two points.

Consider $\omega_-, \xi, \omega_+ \in \operatorname{\mathbb{S}}^1$ and let $\eta \in F^s_\xi$. Since it holds $$F^s_\xi=\left( \overline{F^s_{[[\omega_-,\xi))}} \cap F^s_{\xi} \right) \cup \left( \overline{F^s_{((\xi,\omega_+]]}} \cap F^s_\xi \right)\ ,$$ the claim follows by Lemma \[lemma:one:point\].

We are know ready to prove the main theorem of the section which extends in some sense [@BIW1 Theorem 5.1] to the context of measurable cocycles.

\[teor:boundary:map\] In the situation of Assumption \[ass:zariski:zero:measure\], there exist two measurable maps $$\phi^\pm:\operatorname{\mathbb{S}}^1 \times \Omega \rightarrow \check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}}$$ such that

1.  the slice $\phi^+_s:\operatorname{\mathbb{S}}^1 \rightarrow \check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}}$ is right continuous for almost every $s \in \Omega$;

2.  the slice $\phi^-_s:\operatorname{\mathbb{S}}^1 \rightarrow \check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}}$ is left continuous for almost every $s \in \Omega$;

3.  the maps $\phi^\pm$ are measurable and $\sigma$-equivariant;

4.  for every $\xi \neq \omega $ in $\operatorname{\mathbb{S}}^1$ and almost every $s \in \Omega$, $\phi^\varepsilon_s(\xi)$ is transverse to $\phi^\delta_s(\omega)$, where $\varepsilon, \delta \in \{  \pm \}$;

5.  almost every slice is monotone, that is for every $\xi,\omega,\zeta \in \operatorname{\mathbb{S}}^1$ and almost every $s \in \Omega$ it holds $$\beta_{\operatorname{\mathcal{X}}}(\phi_s^\varepsilon(\xi),\phi_s^\delta(\omega),\phi_s^\theta(\zeta))=\operatorname{rk}(\operatorname{\mathcal{X}}) \beta_{\operatorname{\mathbb{S}}^1}(\xi,\omega,\zeta) \ ,$$ where $\varepsilon,\delta,\theta \in \{ \pm \}$.

Additionally $\phi^\pm$ are essentially unique.

By assumption we know that for almost every $s \in \Omega$, the slice $\phi_s$ is maximal and it satisfies Assumption \[ass:zariski:zero:measure\]. For any such $s$, we define for every $\xi \in \operatorname{\mathbb{S}}^1$ the following maps $$\phi_s^+(\xi)=\overline{F^s_{[[\omega_-,\xi))}} \cap F^s_{\xi}\ , 
\phi_s^-(\xi)=\overline{F^s_{((\xi,\omega_+]]}} \cap F^s_\xi  \ ,$$ where $\omega_- ,\xi,\omega_+$ is a positively oriented triple in $\operatorname{\mathbb{S}}^1$ and $\omega_\pm$ are arbitrary. The right continuity of $\phi^+_s$ and the left continuity of $\phi^-_s$ are clear by their definitions. We can define $$\phi^\pm:\operatorname{\mathbb{S}}^1 \times \Omega \rightarrow \check{\operatorname{\mathcal{S}}}_{\operatorname{\mathcal{X}}}, \ \phi_s^\pm(\xi,s):=\phi_s^\pm(\xi) \ .$$ The measurability of the functions $\phi^\pm_s$ comes from the fact the slice $\phi_s^\pm$ are measurable and varies measurably with respect to $s$ by the measurability of $\phi$. The $\sigma$-equivariance of the latter implies that $\phi^\pm$ are $\sigma$-equivariant.

Finally property $4)$ follows by Lemma \[lemma:transverse\] and property $5)$ follows by Lemma \[lemma:maximality:triples\]. The essential uniqueness is a consequences of the assumption on the essential uniqueness of the boundary map.

[^1]:
