On groups with uncountably many subgroups of finite index

Let K be the kernel of an epimorphism chi:G --> Z, for G a finitely presented group. If K has uncountably many normal subgroups of finite index I, then K has uncountably many subgroups (not necessarily normal) of any finite index greater than r. In particular, this is the case whenever G is subgroup separable and K is nonfinitely generated. Assume that G has an abelian HNN base contained in K. If K has infinitely many subgroups of a given finite index, then it has uncountably many. (C) 1999 Elsevier Science B.V. All rights reserved.