Exponent growth in groups acting on regular rooted trees

Whenever G is a group acting on a regular rooted tree, it is possible to define a canonical chain {Ni} of finite-index normal subgroups of G, which take the name of level stabilizers in G. If moreover G is finitely generated and infinite, it can be proved that the sequence of exponents exp(G/Ni) converges to infinity as i goes to infinity. The problem of determining the speed at which such sequence goes to infinity is called the exponent growth problem for G. In this thesis, after giving an introduction to the general theory of groups acting on regular rooted trees, we will focus on the family of the so-called multi-EGS-groups. This family of groups has been widely studied as a rich source of examples in group theory: it gives rise, for instance, to many examples that answer in the negative the well-known General Burnside Problem. Our final goal will be to solve the exponent growth problem for every multi-EGS-group. We will find an explicit expression for the above sequence of exponents when G is a multi-EGS-group: such expression will turn out to be very simple, it will depend on a few parameters and its growth to infinity will prove to be unexpectedly quick.