On infinitely divisible measures on certain finitely generated groups (Q1319314)

Let \(G\) be a locally compact group. Probability measures on \(G\) form a semigroup \(M^1(G)\) under convolution product. If for a \(\mu\in M^1(G)\) and any natural number \(n\) there exists a \(\nu\in M^1(g)\) such that \(\mu= \nu^n\) then \(\mu\) is said to be infinitely divisible. If there exists a continuous semigroup homomorphism \(\psi: \mathbb{R}_+\to M^1(G)\) such that \(\psi(1)= \mu\) then \(\mu\) is said to be embeddable. Of course, embeddability implies infinite divisibility. The converse is not true in general and if for a group it holds then the group is said to have the embedding property. -- The main goal of the paper under review is to show that the following group \(\Gamma\) has the embedding property: Let \(\Gamma\) be a finitely generated group equipped with the discrete topology. Let \(\mathbb{A}\) be the field of algebraic numbers and suppose that there exists an (abstract) homomorphism \(\Gamma\to \text{GL}(n, \mathbb{A})\) such that \(\text{ker }\psi\) is a finitely generated central subgroup of \(\Gamma\). (In fact even every infinitely divisible measure on \(\Gamma\) is a Poisson measure).