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A construction of the measurable Poisson boundary: from discrete to continuous groups

Let $Γ$ be a dense countable subgroup of a locally compact continuous group $G$. Take a probability measure $μ$ on $Γ$. There are two natural spaces of harmonic functions: the space of $μ$-harmonic functions on the countable group $Γ$ and the space of $μ$-harmonic functions seen as functions on $G$ defined a.s. with respect to its Haar measure $λ$. This leads to two natural Poisson boundaries: the $Γ$-Poisson boundary and the $G$-Poisson boundary. Since boundaries on the countable group are quite well understood, a natural question is to ask how $G$-boundary is related to the $Γ$-boundary. In this paper we present a theoretical setting to build the $G$-Poisson boundary from the $Γ$-boundary. We apply this technics to build the Poisson boundary of the closure of the Baumslag-Solitar group in the group of real matrices. In particular we show that, under moment condition and in the case that the action on $\mathbf{R}$ is not contracting, this boundary is the $p$-solenoid.