Given a group of local biholomorphisms G and a germ of curve, we want to determine whether or not the orbit of the curve is discrete. The topology of the space of curves, in dimension 2, is given by the ultrametric d(C,D) = m(C)m(D)/(C,D), where m(C) is the multiplicity of the curve and (C,D) is the intersection number. As a first approximation to the study of the dynamics of G, it is interesting to study the (simpler) action of G on the space of curves, since such space is essentially a tree.

It was proved by the author that a group G of local biholomorphisms in dimension 2 acts by discrete orbits on the space of curves if and only if the group is finitely determined, meaning that there exists a natural number k such that the elements of G are determined by their kth-jets. Such a result is no longer true in higher dimensions. A natural question arises in dimension higher or equal than 3: How is a non-discrete orbit of curves of an action of a finitely determined group? More precisely, we will characterize such pairs of groups and orbits.