Resumen de Lie algebras and growth in branch groups

Laurent Bartholdi

• We compute the structure of the Lie algebras associated to two examples of branch groups, and show that one has ?nite width while the other, the Gupta¿Sidki group, has unbounded width and Lie algebra of Gelfand¿Kirillov dimension log3/log(1 + ).

  We then draw a general result relating the growth of a branch group, of its Lie algebra, of its graded group ring, and of a natural homogeneous space we call parabolic space, namely the quotient of the group by the stabilizer of an in?nite ray. The growth of the group is bounded from below by the growth of its graded group ring, which connects to the growth of the Lie algebra by a product-sum formula, and the growth of the parabolic space is bounded from below by the growth of the Lie algebra.

  Finally we use this information to explicitly describe the normal subgroups of G, the Grigorchuk group. All normal subgroups are characteristic, and the number bn of normal subgroups of G of index 2n is odd and satis?es limsupbn/nlog 23 = 5log 23, liminfbn/nlog 23 = .