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| Format: | Preprint |
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2020
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| Online Access: | https://arxiv.org/abs/2005.10529 |
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| _version_ | 1866909635267526656 |
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| author | Sarti, Filippo Savini, Alessio |
| author_facet | Sarti, Filippo Savini, Alessio |
| contents | Let $Γ$ be a discrete countable group acting isometrically on a measurable field $\mathbf{X}$ of CAT(0)-spaces of finite telescopic dimension over some ergodic standard Borel probability $Γ$-space $(Ω,μ)$. If $\mathbf{X}$ does not admit any invariant Euclidean subfield, we prove that the measurable field $\widehat{\mathbf{X}}$ extended to a $Γ$-boundary admits an invariant section. In the case of constant fields this shows the existence of Furstenberg maps for measurable cocycles, extending results by Bader, Duchesne and Lécureux.
When $Γ<\mathrm{PU}(n,1)$ is a torsion-free lattice and the CAT(0)-space is $\mathcal{X}(p,\infty)$, we show that a maximal cocycle $σ:Γ\times Ω\rightarrow \mathrm{PU}(p,\infty)$ with a suitable boundary map is finitely reducible. As a consequence, we prove an infinite dimensional rigidity phenomenon for maximal cocycles in $\mathrm{PU}(1,\infty)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2005_10529 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Boundary maps and reducibility for cocycles into the isometries of CAT(0)-spaces Sarti, Filippo Savini, Alessio Geometric Topology 57T10, 53C35, 22E40, 22D40 Let $Γ$ be a discrete countable group acting isometrically on a measurable field $\mathbf{X}$ of CAT(0)-spaces of finite telescopic dimension over some ergodic standard Borel probability $Γ$-space $(Ω,μ)$. If $\mathbf{X}$ does not admit any invariant Euclidean subfield, we prove that the measurable field $\widehat{\mathbf{X}}$ extended to a $Γ$-boundary admits an invariant section. In the case of constant fields this shows the existence of Furstenberg maps for measurable cocycles, extending results by Bader, Duchesne and Lécureux. When $Γ<\mathrm{PU}(n,1)$ is a torsion-free lattice and the CAT(0)-space is $\mathcal{X}(p,\infty)$, we show that a maximal cocycle $σ:Γ\times Ω\rightarrow \mathrm{PU}(p,\infty)$ with a suitable boundary map is finitely reducible. As a consequence, we prove an infinite dimensional rigidity phenomenon for maximal cocycles in $\mathrm{PU}(1,\infty)$. |
| title | Boundary maps and reducibility for cocycles into the isometries of CAT(0)-spaces |
| topic | Geometric Topology 57T10, 53C35, 22E40, 22D40 |
| url | https://arxiv.org/abs/2005.10529 |