Invariable generation does not pass to finite index subgroups

Abstract

Using small cancellation methods, we show that invariable generation does not pass to finite index subgroups, answering questions of Wiegold (1977) and Kantor–Lubotzky–Shalev (2015). We further show that a finitely generated group that is invariably generated is not necessarily finitely invariably generated, answering a question of Cox (2021). The same results were also obtained independently by Minasyan (2021).