Finiteness conditions in groups of tree automorphisms

For any prime p and for every natural c > 2, we construct an n-generated (n-2)-finite residually finite p-group of unbounded period such that the growth function of periods of its elements grows slower than any number of logarithms does. This gives the answer to Grigorchuk's question. Also constructed are residually finite but not locally finite p-groups, in which every finite set of elements whose order does not exceed a fixed one generate a finite subgroup. In the class of residually finite groups, some of the finiteness conditions specified by Shunkov are generalized and separated. In particular, question 8.66 from the Kourovka Notebook is settled and some progress is made toward solving problems 9.77 and 9.78. © 1998 Plenum Publishing Corporation.