Lie Algebras and Growth in Branch Groups

Abstract

We compute the structure of the Lie algebras associated to two examples of branch groups, and show that one has finite width while the other, the ``Gupta-Sidki group'', has unbounded width. This answers a question by Sidki. More precisely, the Lie algebra of the Gupta-Sidki group has Gelfand-Kirillov dimension $\log3/\log(1+\sqrt2)$. 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 infinite 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 the "Grigorchuk group". All normal subgroups are characteristic, and the number of normal subgroups of index $2^n$ is odd and is asymptotically $n^{\log_2(3)}$.