A finitely generated residually finite group with an unsolvable word problem

Abstract

The group described in the title is obtained as a quotient of a center-by-metabelian group constructed by P. Hall. It is well known that a residually finite group which is finitely presented has a solvable word problem (V. H. Dyson [1], A. W. Mostowski [6]). It is also known (see the last sentence in V. H. Dyson [1]) although perhaps not well enough (see W. Magnus [5, p. 307]) that a residually finite group which is only finitely generated may have an unsolvable word problem. Herein we exhibit such a group which moreover is soluble, indeed it is center-by-metabelian. This is in some sense a minimal soluble example since all finitely generated metabelian groups have a solvable word problem. (This is an easy consequence of the fact that finitely generated free metabelian groups satisfy the maximal condition on normal subgroups (P. Hall [2]) and thus any finitely generated metabelian group can be almost finitely presented in the sense of Mostowski [6] and is residually finite (P. Hall [3]).) Credit must be given to Peter Neumann who pointed out the appropriate example to be modified. Our example uses a recursive function f with a nonrecursive range in a manner similar to the way Higman [5] exhibited a finitely presented group with an unsolvable word problem. Let G be the center-by-metabelian group constructed by P. Hall in ([2, p. 434]). In the notation of P. Hall, G is generated by a and b and defined by [bi, bj, bk] = 1 where bi = a-ibai, i, j, k = 0, 1, ... Ci; = Ci+kj+ where cij = [bj, bi], j > i, i, j, k = 0, ?1, The center C of G is free abelian with basis dl, d2, i i where dr = cii+, r = 1,2 ,i= 0, ?1, *. Received by the editors March 12, 1973. AMS (MOS) subject classifications (1970). Primary 20E25, 20F10; Secondary 20E1 5.