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 index ≤ 212 in the Grigorchuk group G is given. This leads to the following sharp version of the congruence property: a normal subgroup not contained in the stabilizer at level n + 1 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 determination of all normal subgroups at each level n ≥ 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 automorphism group of the regular rooted tree.