Polynomially growing harmonic functions on connected groups

We study the connection between dimension of certain spaces harmonic functions on a group and its geometric algebraic properties. Our main result shows that (for sufficiently “nice” random walk measures) connected, compactly generated, locally compact has polynomial volume growth if only space linear finite dimension. This characterization is interesting in light fact Gromov’s theorem regarding finitely generated groups does not have an analog connected case. That is, there are examples nilpotent by compact. Also, analogous for discrete case been established solvable still open general groups.