On systems of equations in a free group. (English. Russian original) Zbl 0579.20019

Math. USSR, Izv. 25, 115-162 (1985); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 48, No. 4, 779-832 (1984).
In this rather long paper the author deals with the general solution of given bounded periodicity exponent of an arbitrary system of equations in a free group. The periodicity exponent of a list of words \(A_ 1,...,A_ K\) is defined to be the maximal positive integer s such that some word of this list contains a subword of the form \(B^ s\) for some nonempty word B. It is difficult to give sufficient details of the contents of this paper since even the definitions will occupy considerable space. I shall give here without detailed explanations the main result: For a given system of equations \[ \phi_ 1(x_{i_ 1},...,x_{i_ n})=1,...,\phi_ m(x_{i_ 1},...,x_{i_ n})=1 \] in a free group and a positive integer s it is possible to effectively construct a finite set of fundamental sequences for this system in such a way that each solution, of the system \(\phi_ 1=1,...,\phi_ m=1\), with periodicity exponent at most s is described by at least one of the sequences.
Reviewer: S.Andreadakis


20E05 Free nonabelian groups
20F05 Generators, relations, and presentations of groups
20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
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