The conjugacy problem for certain branch groups (Q2783053)

In 1980, the first author of this paper constructed and studied some remarkable new examples of finitely generated torsion groups. These groups are residually finite and amenable \(p\)-groups. Five years later, he solved a problem of Milnor by showing that their growth is faster than polynomial growth and slower than exponential growth. In 1983, a similar construction of 2-generated infinite \(p\)-groups was obtained by Gupta and Sidki. These two constructions and their variations give a big class of infinite groups called branch groups.NEWLINENEWLINENEWLINEThe purpose of the paper is to solve the conjugacy problem in a uniform way for a wide class of branch groups under some weak conditions on the length function. This class contains Grigorchuk and Gupta-Sidki \(p\)-groups (\(p>2\)) for which the conjugacy problem was solved by \textit{J. S. Wilson} and the reviewer [in Arch. Math. 68, No. 6, 441-449 (1997; Zbl 0877.20020)] as well as Grigorchuk's group for \(p=2\) for which the conjugacy problem was independently solved by \textit{Yu. G. Leonov} [Mat. Zametki 64, No. 4, 573-583 (1998; Zbl 0942.20011)] and by \textit{A. V. Rozhkov} [ibid. 64, No. 4, 592-597 (1998; Zbl 0949.20025)]. The class also contains some torsion-free examples. The advantage of the authors' approach is that in many cases it leads also to the solubility of the conjugacy problem for subgroups of finite index.NEWLINENEWLINEFor the entire collection see [Zbl 0981.00006].