Poisson boundary for finitely generated groups of rational affinities

Abstract

The group of affine transformations with rational coefficients, $aff(Q)$, acts naturally on the real line, but also on the $p$-adic fields. The aim of this note is to show that all these actions are necessary and sufficient to represent bounded $\mu$-harmonic functions for a probability measure $\mu$ on $aff(Q)$ that is supported by a finitely generated sub-group, that is to describe the Poisson boundary.