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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.