The Milnor-Wolf theorem states that any *finitely generated* solvable group has either polynomial or exponential growth. 



> 
> Is there an analogous result for locally compact compactly generated groups?
>  (or rather, for some smaller class of group? connected ones? connected Lie?)
> 
> 
> 



Milnor-Wolf theorem for topological groups
I found the following paper:


Yves Guivarc'h, Croissance polynomiale et périodes des fonctions harmoniques, Bulletin de la Société Mathématique de France (1973) Volume: 101, page 333-379


The analogous result (see Corollaire III.3) is proved for all compactly generated soluble locally compact groups as well as some other cases. Here 'growth' is measured in terms of Haar measure, i.e. the asymptotic growth rate of $\mu(U^n)$ where $U$ is some compact generating set with nonempty interior.


