# Averaged projections, angles between groups and strengthening of Banach property (T)

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## Abstract

Recently, Lafforgue introduced a new strengthening of Banach property (T), which he called strong Banach property (T) and showed that this property has implications regarding fixed point properties and Banach expanders. In this paper, we introduce a new strengthening of Banach property (T), called “robust Banach property (T)”, which is weaker than strong Banach property (T), but is still strong enough to ensure similar applications. Using the method of averaged projections in Banach spaces and introducing a new notion of angles between projections, we establish a criterion for robust Banach property (T) and show several examples of groups in which this criterion is fulfilled. We also derive several applications regarding fixed point properties and Banach expanders and give examples of these applications.

## Mathematics Subject Classification

20F99 46B85## Notes

### Acknowledgments

The author would like to thank Mikael de la Salle, Mikhail Ershov and Simeon Reich for reading a previous draft of this paper and adding their insightful comments. Most of the work was done while the author was a visiting assistant professor at the Ohio State University and the author thanks the University for its hospitality.

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