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This is clear for D-width, because the discrete and pro-p Lie algebras £(G) and C{G) are isomorphic. The finiteness of C-width follows from the inequalities 7n(G)/7»+i(G)| < |7n(G)/ 7n+ i(G)| < oo, which again are consequences of the congruence property of G. 1: it is neither solvable, because G isn't, nor p-adic analytic, by Lazard's criterion [Laz65] (its Lie algebra C(G) = C{G) would have a zero component in some dimension). 1 are hereditarily just-infinite groups, that is, groups every open subgroup of which is just-infinite [KLP97, page 5].

1. The relation < is a linear quasiorder, and s < t holds whenever s is a proper subterm oft. We can easily verify that u < u holds for all u e W(S). The rest of the proof will be divided into five steps. Step 1. s < t holds if s is a subterm oft. Proof. Proceed by induction on \s\ + \t\. We clearly have s < t if s = t or s is an argument of t. In other cases one can express t as Ft\.. tn, with s a subterm of tj, 1 < j < n. Then s < tj by the induction assumption, and s < t by (1). Step 2. Ifv

Publ. Math. (1981), no. 53, 53-73. [Gro81bJ Mikhael Gromov, Structures metriques pour les varietes riemanniennes, CEDIC, Paris, 1981, Edited by J. Lafontaine and P. Pansu. [Gui70] Yves Guivarc'h, Groupes de Lie a croissance polynomial, C. R. Acad. Sci. Paris Ser. A-B 271 (1970), A237-A239. [Gui73] Yves Guivarc'h, Croissance polynomiale et pe'riodes des fonctions harmoniques, Bull. Soc. Math. France 101 (1973), 333-379. [GS83] Narain Gupta and Said Sidki, On the Burnside problem for periodic groups, Math.