The Top of the Lattice of Normal Subgroups of the Grigorchuk Group

AbstractA 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