A construction which can be used to produce finitely presented infinite simple groups
This is the first in a series of three papers discussing finitely presented simple groups. The work is based on earlier work by \textit{R. J. Thompson} [Word problems II, Stud. Logic Found. Math. 95, 401-441 (1980; Zbl 0431.20030)] and \textit{G. Higman} [''Finitely presented simple groups'', Notes on Pure Mathematics 8, The Australian National University Canberra (1974)]. The aim is to describe a procedure which will yield a large class of finitely presented simple groups. Let W be a free monoid of rank n greater than 1. A right ideal Y of W is called inescapable if, given any u in W, there is some w in W with uw in Y. The ideal Y is called cofinite if Y is cofinite in W as subset. A function \(\mu\) on a subspace of W is called a homomorphism if \((uw)\mu =(u\mu)w\) whenever these are defined, \(\mu\) is an isomorphism if it is also bijective between its domain and co-domain. Further, \(\mu\) is inescapable (cofinite) it its domain and range are inescapable (cofinite). It is not difficult to show that every inescapable isomorphism has a unique maximal inescapable extension. The inescapable isomorphisms then form a group \({\mathcal G}_{n,1}\) with the operation being ''composition followed by formation of the maximal extension''. Similarly the cofinite isomorphisms form a subgroup \(G_{n,1}\). (The latter is amongst the groups constructed by Higman in [loc. cit].) The author constructs her groups by taking suitable subgroups H (obtained as subgroups of inverse limits of wreath products) of \({\mathcal G}_{n,1}\) and showing that the groups \(<G_{n,1},H>\) are finitely presented. She then uses a result, which she ascribes largely to Thompson that if K is a subgroup of \({\mathcal G}_{n,1}\) containing \(G_{n,1}\) then the derived group of K is simple. Since the finitely presented groups mentioned above have derived group of finite index, this yields the required class of finitely presented simple groups.
scientific article; zbMATH DE number 3865565
scientific article