  The growth of a finitely generated group is an important geometric invariant
which has been studied for decades. It can be either polynomial, for a
well-understood class of groups, or exponential, for most groups studied by
geometers, or intermediate, that is between polynomial and exponential. Despite
recent spectacular progresses, the class of groups with intermediate growth
remains largely mysterious. Many examples of such groups are constructed using
Mealy automata. The aim of this paper is to give an algorithmic procedure to
study the growth of such automata groups, and more precisely to provide
numerical upper bounds on their exponents. Our functions retrieve known optimal
bounds on the famous first Grigorchuk group. They also improve known upper
bounds on other automata groups and permitted us to discover several new
examples of automata groups of intermediate growth. All the algorithms
described are implemented in GAP, a language dedicated to computational group
theory.
