possible bridge between group growth theory and complexity theory?
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.