Lie algebras and growth in branch groups

Laurent Bartholdi

Vol. 218 (2005), No. 2, 241–282

DOI: 10.2140/pjm.2005.218.241

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 and Lie algebra of Gelfand–Kirillov dimension log 3∕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 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 G, the Grigorchuk group. All normal subgroups are characteristic, and the number b_n of normal subgroups of G of index 2ⁿ is odd and satisfies limsupb_n∕n^{log ₂3} = 5^{log ₂3}, liminf b_n∕n^{log ₂3} = .

Keywords

Lie algebra, growth of groups, lower central series

Mathematical Subject Classification 2000

Primary: 20F14, 20F40, 17B70, 16P90, 20E08

Milestones

Received: 23 April 2002

Revised: 10 October 2003

Published: 1 February 2005