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On the trivializability of rank-one cocycles with an invariant field of projective measures

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Abstract

Let G be \(SO ^\circ (n,1)\) for \(n \geqslant 3\) and consider a lattice \(\Gamma < G\). Given a standard Borel probability \(\Gamma \)-space \((\Omega ,\mu )\), consider a measurable cocycle \(\sigma :\Gamma \hspace{1.111pt}{\times }\hspace{1.111pt}\Omega \rightarrow {\textbf{H}}(\kappa )\), where \({\textbf{H}}\) is a connected algebraic \(\kappa \)-group over a local field \(\kappa \). Under the assumption of compatibility between G and the pair \(({\textbf{H}},\kappa )\), we show that if \(\sigma \) admits an equivariant field of probability measures on a suitable projective space, then \(\sigma \) is trivializable. An analogous result holds in the complex hyperbolic case.

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Acknowledgements

The author would like to thank the anonymous referee for the precious suggestions and comments which allowed him to improve the quality of the paper.

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  1. Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133, Milan, Italy

    Alessio Savini

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Savini, A. On the trivializability of rank-one cocycles with an invariant field of projective measures. European Journal of Mathematics 10, 8 (2024). https://doi.org/10.1007/s40879-023-00721-1

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