On groups with uncountably many subgroups of finite index (Q1300654)

It is well known that every finitely generated group has only finitely many subgroups of a given finite index. The authors consider groups \(K\) which are close to finitely generated in the following sense; namely, \(K\) is a normal subgroup of some finitely presented group \(G\) with \(G/K\) infinite cyclic. They prove (Theorem 1.2) that if \(K\) has uncountably many normal subgroups of finite index \(r\) then \(K\) has uncountably many (not necessarily normal) subgroups of every finite index \(\geq r\). The hypothesis ``finitely presented'' cannot be replaced by ``finitely generated''. They also show (Corollary 3.4) that when \(K\) contains an Abelian HNN base for \(G\), and \(K\) has infinitely many subgroups of a fixed finite index \(r\), then \(K\) has uncountably many subgroups of this index. This leads to the theorem (Theorem 4.1) that, if \(K\) is endowed with the profinite topology inherited from \(G\), then \(K\) has only countably many subgroups of finite index if and only if \(K\) is topologically finitely generated.