On the growth of actions of free products

If G is a finitely generated group and X a G -set, the growth of the action of G on X is the function that measures the largest cardinality of a ball of radius n in the (possibly non-connected) Schreier graph (G, X). We consider the following stability problem: if G, H are finitely generated groups admitting a faithful action of growth bounded above by a function f, does the free product G H also admit a faithful action of growth bounded above by f? We show that the answer is positive under additional assumptions, and negative in general. In the negative direction, our counter-examples are obtained with G either the commutator subgroup of the topological full group of a minimal and expansive homeomorphism of the Cantor space, or G a Houghton group. In both cases, the group G admits a faithful action of linear growth, and we show that G H admits no faithful action of subquadratic growth provided H is non-trivial. In the positive direction, we describe a class of groups that admit actions of linear growth and is closed under free products and exhibit examples within this class, among which the Grigorchuk group.