# Milnor-Wolf theorem for topological groups

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?)

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.
• Also the type of growth can be characterized with no reference to the Haar measure. Let $K$ be a compact symmetric generating neighborhood of 1 in $G$. Let $X$ be a maximal subset for the property that $x,y\in X$, $x\neq y$ implies $x^{-1}y\notin K$. Make $X$ a graph saying $x-y$ if $x^{-1}y\in K^3$. Then this is a connected graph of bounded degree, and its type of growth (up to the usual asymptotic equivalence) does not depend on the choices, and indeed is the same as that obtained with the Haar measure. – YCor Mar 8 '17 at 13:14