首页/论文/2002.03628

arXiv 2002.036282020-02math.GT

Superrigidity of maximal measurable cocycles of complex hyperbolic lattices

Filippo Sarti, Alessio Savini

备注：: Commenta are welcome! 25 pages
DOI：: 10.1007/s00209-021-02801-y

英文摘要：

Let $Γ$ be a torsion-free lattice of $\text{PU}(p,1)$ with $p \geq 2$ and let $(X,μ_X)$ be an ergodic standard Borel probability $Γ$-space. We prove that any maximal Zariski dense measurable cocycle $σ: Γ\times X \longrightarrow \text{SU}(m,n)$ is cohomologous to a cocycle associated to a representation of $\text{PU}(p,1)$ into $\text{SU}(m,n)$, with $1 < m \leq n$. The proof follows the line of Zimmer' Superrigidity Theorem and requires the existence of a boundary map, that we prove in a much more general setting. As a consequence of our result, it cannot exist a maximal measurable cocycle with the above properties when $n\neq m$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Superrigidity from Chevalley groups into acylindrically hyperbolic groups via quasi-cocycles

We prove that every homomorphism from the elementary Chevalley group over a finitely generated unital commutative ring associated with reduced irreducible classical root system of rank at least 2, and ME analogues of such groups, into acylindrically hyperbolic groups has an absolutely elliptic image. This result provides a non-arithmetic generalization of homomorphism superrigidity of Farb--Kaimanovich--Masur and Bridson--Wade.

2018-01-26math.GRmath.GTmath.KT

Cocycle superrigidity from higher rank lattices to $\mathrm{Out}(F_N)$

We prove a rigidity result for cocycles from higher rank lattices to $\mathrm{Out}(F_N)$ and more generally to the outer automorphism group of a torsion-free hyperbolic group. More precisely, let $G$ be either a product of connected higher rank simple algebraic groups over local fields, or a lattice in such a product. Let $G\curvearrowright X$ be an ergodic measure-preserving action on a standard probability space, and let $H$ be a torsion-free hyperbolic group. We prove that every Borel cocycle $G\times X\to\mathrm{Out}(H)$ is cohomologous to a cocycle with values in a finite subgroup of $\ma…

2022-10-11math.GRmath.GT

Borel invariant for measurable cocycles of 3-manifold groups

We introduce the notion of pullback along a measurable cocycle and we use it to extend the Borel invariant studied by Bucher, Burger and Iozzi to the world of measurable cocycles. The Borel invariant is constant along cohomology classes and has bounded absolute value. This allows to define maximal cocycles. We conclude by proving that maximal cocycles are actually trivializable to the restriction of the irreducible representation.

2021-12-30math.GT

Multiplicative constants and maximal measurable cocycles in bounded cohomology

Multiplicative constants are a fundamental tool in the study of maximal representations. In this paper we show how to extend such notion, and the associated framework, to measurable cocycles theory. As an application of this approach, we define and study the Cartan invariant for measurable $\textup{PU}(m,1)$-cocycles of complex hyperbolic lattices.

2021-07-21math.GT

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

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 $\mathbb{R}$-group such that $G=\mathbf{G}(\mathbb{R})^\circ$ is of Hermitian type. If $Γ\leq L$ is a torsion-free lattice of a finite connected covering of $\text{PU}(1,1)$, given a standard Borel probability $Γ$-space $(Ω,μ_Ω)$, we introduce the notion of Toledo invariant for a measurable cocycle $σ:Γ\times Ω\rightarrow G$. The Toledo remains unchanged along $G$-cohomology clas…

2020-11-20math.GT

Equivariant maps for measurable cocycles with values into higher rank Lie groups

Let $G$ a semisimple Lie group of non-compact type and let $\mathcal{X}_G$ be the Riemannian symmetric space associated to it. Suppose $\mathcal{X}_G$ has dimension $n$ and it has no factor isometric to either $\mathbb{H}^2$ or $\text{SL}(3,\mathbb{R})/\text{SO}(3)$. Given a closed $n$-dimensional Riemannian manifold $N$, let $Γ=π_1(N)$ be its fundamental group and $Y$ its universal cover. Consider a representation $ρ:Γ\rightarrow G$ with a measurable $ρ$-equivariant map $ψ:Y \rightarrow \mathcal{X}_G$. Connell-Farb described a way to construct a map $F:Y\rightarrow \mathcal{X}_G$ which is smo…

2021-04-30math.GT

更多相关论文收起更多论文

Parametrized Euler class and semicohomology theory

We extend Ghys' theory about semiconjugacy to the world of measurable cocycles. More precisely, given a measurable cocycle with values into $\text{Homeo}^+(\mathbb{S}^1)$, we can construct a $\text{L}^\infty$-parametrized Euler class in bounded cohomology. We show that such a class vanishes if and only if the cocycle can be lifted to $\text{Homeo}^+_{\mathbb{Z}}(\mathbb{R})$ and it admits an equivariant family of points. We define the notion of semicohomologous cocycles and we show that two measurable cocycles are semicohomologous if and only if they induce the same parametrized Euler class. S…

2021-01-28math.GT

多项式映射与上循环的自动连续性

Automatic continuity of Polynomial maps and cocycles

本文推广经典同态自动连续性定理，证明实数上被d+1个差分算子零化的Lebesgue可测函数必为次数至多d的多项式，并进一步建立局部紧群间Haar可测多项式映射的连续性，其核心工具是上循环的自动连续性定理。

2024-08-29math.GT

CAT(0)-空间等距群的余循环的边界映射与可约性

Boundary maps and reducibility for cocycles into the isometries of CAT(0)-spaces

本文研究离散群在CAT(0)-空间可测场上的等距作用，证明无不变欧几里得子场时延拓到边界存在不变截面，并由此得到极大余循环的有限可约性与无穷维刚性现象。

2025-06-04math.GT

A Matsumoto-Mostow result for Zimmer's cocycles of hyperbolic lattices

As for the theory of maximal representations, we introduce the volume of a Zimmer's cocycle $Γ\times X \rightarrow \mbox{PO}^\circ(n, 1)$, where $Γ$ is a torsion-free (non-)uniform lattice in $\mbox{PO}^\circ(n, 1)$, with $n \geq 3$, 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 …

2020-09-24math.GT

Natural maps for measurable cocycles of compact hyperbolic manifolds

Let $\text{G}(n)$ be equal either to $\text{PO}(n,1),\text{PU}(n,1)$ or $\text{PSp}(n,1)$ and let $Γ\leq \text{G}(n)$ be a uniform lattice. Denote by $\mathbb{H}^n_K$ the hyperbolic space associated to $\text{G}(n)$, where $K$ is a division algebra over the reals of dimension $d=\dim_{\mathbb{R}} K$. Assume $d(n-1) \geq 2$. In this paper we generalize natural maps to measurable cocycles. Given a standard Borel probability $Γ$-space $(X,μ_X)$, we assume that a measurable cocycle $σ:Γ\times X \rightarrow \text{G}(m)$ admits an essentially unique boundary map $ϕ:\partial_\infty \mathbb{H}^n_K \ti…

2021-09-02math.DGmath.GT

参数化Kähler类与Zariski稠密轨道1-上同调

Parametrized Kähler class and Zariski dense orbital 1-cohomology

本文在遍历群作用背景下引入参数化Kähler类，证明其可完全刻画Zariski稠密可测上闭链直至上同调等价。

2025-06-04math.GT

Semisimplicity and rigidity of the Kontsevich-Zorich cocycle

We prove that invariant subbundles of the Kontsevich-Zorich cocycle respect the Hodge structure. In particular, we establish a version of Deligne semisimplicity in this context. This implies that invariant subbundles must vary polynomially on affine manifolds. All results apply to tensor powers of the cocycle and this implies that the measurable and real-analytic algebraic hulls coincide. We also prove that affine manifolds parametrize Jacobians with non-trivial endomorphisms. Typically a factor has real multiplication. The tools involve curvature properties of the Hodge bundles and estimates …

2014-08-04math.AGmath.DSmath.GT