Lie methods in growth of groups and groups of finite width (Q2702042)

The paper gives a method of proving that some groups from the class of branch groups have finite width. The authors explicitly compute the lower central series and the dimension series of two branch groups, a Grigorchuk group [\textit{R. I. Grigorchuk}, Funkts. Anal. Prilozh. 14, No. 1, 53-54 (1980; Zbl 0595.20029)] and another one, containing a Grigorchuk group as a subgroup, and prove that both have finite width. The profinite completions of these groups represent counterexamples to Zelmanov's conjecture on the structure of just-infinite pro-\(p\)-groups of finite width. Associating to these groups Lie algebras provides new examples of Lie algebras of finite width.NEWLINENEWLINENEWLINELie methods in growth of groups are also discussed. Combining Lie methods with the Golod-Shafarevich construction, the authors produce examples of finitely generated \(p\)-groups of uniformly exponential growth.NEWLINENEWLINEFor the entire collection see [Zbl 0940.00028].