The top of the lattice of normal subgroups of the Grigorchuk group

A complete description of the lattice of all normal subgroups not contained in the stabilizer of the fourth level of the tree and, consequently, of in dex less than or equal to 2(12) in the Grigorchuk group G is given. This le ads to the following sharp version of the congruence property: a normal sub group not contained in the stabilizer at level n + contains the stabilizer at level n + 3 (in fact such a normal subgroup contains the subgroup Nn+1), but, in general, it does not contain the stabilizer at level n + 2. The de termination of all normal subgroups at each level n greater than or equal t o 4 is then reduced to the analysis of certain G-modules which depend only on n and the previous description, as for the analogous problem for the aut omorphism group of the regular rooted tree. (C) 2001 Elsevier Science.