Infinite dimensional non-positively curved symmetric spaces of finite rank

TL;DR

This paper investigates infinite-dimensional symmetric spaces of finite rank, demonstrating they have finite telescopic dimension and establishing the existence of Furstenberg maps, advancing superrigidity theory in infinite dimensions.

Contribution

It introduces the concept of finite telescopic dimension for these spaces and proves the existence of Furstenberg maps, extending superrigidity results to infinite-dimensional settings.

Findings

Spaces have finite telescopic dimension

Existence of Furstenberg maps for certain group actions

Progress towards superrigidity in infinite dimensions

Abstract

This paper concerns a study of three families of non-compact type symmetric spaces of infinite dimension. Although they have infinite dimension they have finite rank. More precisely, we show they have finite telescopic dimension. We also show the existence of Furstenberg maps for some group actions on these spaces. Such maps appear as a first step toward superrigidity results.