A group of non-uniform exponential growth

Laurent Bartholdi

I will give a simple construction of a group W of non-uniform exponential growth. This means that the volume of balls in W's Cayley graph grows exponentially as a function of the radius; but the infimum of these growth rates over all word metrics is 1.

The construction also yields a group V of intermediate growth between polynomial and exponential, that is a "limit" of W in the sense that arbitrarily large balls in V can be embedded metrically in W for an appropriate word metric in W.

The groups V and W are defined by their action on a rooted tree, and are generated by automata.

John Wilson announced in 2002 the existence of a group of non-uniform exponential growth, answering a question by Mikhael Gromov. My construction follows essentially the same lines as Wilson's, but is much shorter.