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Lie Methods in Growth of Groups and Groups of Finite Width

Presented at Edinburgh, 1998
PublisherCambridge : Cambridge University Press
  • London Mathematical Society lecture note series; 275
Publication date2000

In the first, mostly expository, part of this paper, a graded Lie algebra is associated to every group G given with an N-series of subgroups. The asymptotics of the Poincare series of this algebra give estimates on the growth of the group G. This establishes the existence of a gap between polynomial growth and growth of type $e^{sqrt n}$ in the class of residually-p groups, and gives examples of finitely generated p-groups of uniformly exponential growth. In the second part, we produce two examples of groups of finite width and describe their Lie algebras, introducing a notion of Cayley graph for graded Lie algebras. We compute explicitly their lower central and dimensional series, and outline a general method applicable to some other groups from the class of branch groups. These examples produce counterexamples to a conjecture on the structure of just-infinite groups of finite width.

  • arxiv : math.GR
Citation (ISO format)
BARTHOLDI, Laurent, GRIGORCHUK, Rostislav. Lie Methods in Growth of Groups and Groups of Finite Width. In: Computational and geometric aspects of modern algebra. Edinburgh. Cambridge : Cambridge University Press, 2000. (London Mathematical Society lecture note series)
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