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Conjecturably, prop groups with this spectrum are virtually procyclic (it suffices to show that they are p-adic analytic). Some characterization of p-adic analytic groups in terms of Hausdorff dimension can nevertheless be obtained. 2 that infinite subgroups of a p-adic analytic pro- p group have positive Hausdorff dimension (since they have positive dimension as Lie groups). It is shown in [12] that this property characterizes p-adic analytic groups. 4. (Barnea-Shalev, 1997) The following are equivalent for a finitely generated pro-p group G.

Tm ]] with m ::: 1. The concept of R-analytic group is defined in [27], Chapter 13, where it is shown that every R-analytic group has an open subgroup H which is a so-called standard group: that is, the underlying set of H may be identified with (mh)(d), the group operation being given by a formal group law (see [43]); here, h ::: 1 is called the level and d is the dimension of H. • Fd(X, y» for x. y E H = (mh) (d). Let B i (X. Y) be the sum of all terms of Fi (X. Y) having total degree 1 in the X's and total degree 1 in the Y's.

Note that the above construction is in a way the inverse of the one described in our sketch of the proof of Conjecture C. There we passed from an infinite-dimensional Lie algebra of finite coclass to a finite-dimensional perfect Lie algebra admitting a certain non-singular derivation which cannot exist. Here we pass from a finite-dimensional perfect Lie algebra with a certain non-singular derivation to an infinite-dimensional Lie algebra of finite coclass. For a systematic study of non-singular derivations of finite-dimensional simple modular Lie algebras of Cartan type, see the work [14] by Benkart, Kostrikin and Kuznetsov.

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Annemann's mental bargain effects (eleven miracles of mentalism) by Theo Annemann


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