A system of defining relations for a Grigorchuk group
The author exhibits a presentation for a certain 2-group of infinite exponent. The group G was constructed by \textit{R. I. Grigorchuk} [Funkts. Anal. Prilozh. 14, 53-54 (1980; reviewed above)]: it is generated by three permutations, of order two, of the set of reals in the unit interval excluding rationals whose denominators are powers of 2. In later work \textit{R. I. Grigorchuk} has shown [Izv. Akad. Nauk SSR, Ser. Mat. 48, No.5, 939-985 (1984; Zbl 0583.20023)] (the reference in the paper under review is inaccurate) that G is not finitely presentable. The author gives the following recursive procedure for defining a sequence \(w_ n\) (n\(\geq 0)\) of words on four letters a,b,c,d: \(w_ 0=ad\) and for \(i\geq 0\) \(w_{i+1}\) is obtained from \(w_ i\) by the substitutions \(a\to aca\), \(b\to d\), \(c\to b\), \(d\to c\). He then proves, via a sequence of lemmas, that G is generated by a,b,c, and d subject to the relations \(a^ 2=b^ 2=c^ 2=d^ 2=bcd=1=w^ 4_ n=(w_ nw_{n+1})^ 4\), \(n\geq 0\).
scientific article; zbMATH DE number 3957422