Polynomially growing harmonic functions on connected groups

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