# Furstenberg boundaries for pairs of groups

## TL;DR

This paper explores the concept of Furstenberg boundaries for pairs of groups, extending the classical notion to include subgroup relations and examining their properties and universal objects.

## Contribution

It introduces and analyzes the relative Furstenberg boundary $oundary(G,H)$ for pairs of groups, highlighting cases where universality holds or fails.

## Key findings

- Determines the boundary $oundary(G,H)$ in specific cases.
- Shows universality does not always hold for discrete groups.
- Identifies properties of boundaries that may be unexpected.

## Abstract

Furstenberg has associated to every topological group $G$ a universal boundary $\partial(G)$. If we consider in addition a subgroup $H<G$, the relative notion of $(G,H)$-boundaries admits again a maximal object $\partial(G,H)$. In the case of discrete groups, an equivalent notion was introduced by Bearden--Kalantar as a very special instance of their constructions. However, the analogous universality does not always hold, even for discrete groups. On the other hand, it does hold in the affine reformulation in terms of convex compact sets, which admits a universal simplex $\Delta(G,H)$, namely the simplex of measures on $\partial(G,H)$. We determine the boundary $\partial(G,H)$ in a number of cases, highlighting properties that might appear unexpected.