Let Γ be a discrete countable group acting isometrically on a measurable field X of CAT(0)-spaces of finite telescopic dimension over some ergodic standard Borel probability Γ-space (Ω,μ). If X does not admit any invariant Euclidean subfield, we prove that the measurable field 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<PU(n,1) is a torsion-free lattice and the CAT(0)-space is X(p,∞), we show that a maximal cocycle σ:Γ×Ω→PU(p,∞) with a suitable boundary map is finitely reducible. As a consequence, we prove an infinite-dimensional rigidity phenomenon for maximal cocycles in PU(1,∞).
1
University of Pisa, Italy
2
University of Milano-Bicocca, Italy
Filippo Sarti; Alessio Savini. Boundary maps and reducibility for cocycles into the isometries of CAT(0)-spaces. Groups, geometry, and dynamics, Tome 19 (2025) no. 3, pp. 1013-1040. doi: 10.4171/ggd/909
@article{10_4171_ggd_909,
author = {Filippo Sarti and Alessio Savini},
title = {Boundary maps and reducibility for cocycles into~the~isometries of {CAT(0)-spaces}},
journal = {Groups, geometry, and dynamics},
pages = {1013--1040},
year = {2025},
volume = {19},
number = {3},
doi = {10.4171/ggd/909},
url = {http://geodesic.mathdoc.fr/articles/10.4171/ggd/909/}
}
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AU - Alessio Savini
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