possible bridge between group growth theory and complexity theory? RJ Lipton conjectures a link between group growth theory and complexity theory. Group growth theory has undergone rapid advance in the last decade and has many surface similarities/ parallels with complexity theory. (of course there are other deep known links between TCS/ group theory.)


  has anyone seen a link between group growth theory and complexity theory? eg have any complexity class separations been found expressible in this framework, etc?


(eg conceivably there could be an intermediate link via boolean circuits etc.)
Maybe this is along the lines you are looking for: I wrote a blog post here explaining how you can use Gromov's theorem on groups of polynomial growth to show that non-uniform read once automata are no more powerful than linear time Turing machines for deciding word problems of groups.

The basic idea is that groups with non-uniform read once automata necessarily have polynomial growth (i.e., the number of elements with a word of length at most $r$ is $O(r^k)$ for some fixed $k$). By Gromov's theorem, this implies that the group is virtually nilpotent. There is work giving linear time algorithms for the word problems of such groups, so that's that.