# Furstenberg boundary of minimal actions

## TL;DR

This paper generalizes Furstenberg boundaries to minimal group actions, providing characterizations and applications to crossed product exactness, thereby addressing an open problem in the theory of group boundaries.

## Contribution

It introduces the concept of ({b5}, X)-boundary for minimal actions, offers new characterizations, and applies these results to determine conditions for exactness of reduced crossed products.

## Key findings

- Characterization of ({b5}, X)-boundaries via essential and proximal extensions
- Negative resolution of a problem posed by Hadwin and Paulsen
- Necessary and sufficient conditions for reduced crossed product exactness

## Abstract

For a countable discrete group {\Gamma} and a minimal {\Gamma}-space X, we study the notion of ({\Gamma}, X)-boundary, which is a natural generalization of the notion of topological {\Gamma}-boundary in the sense of Furstenberg. We give characterizations of the ({\Gamma}, X)-boundary in terms of essential or proximal extensions. The characterization is used to answer a problem of Hadwin and Paulsen in negative. As an application, we find necessary and sufficient condition for the corresponding reduced crossed product to be exact.