Invariable generation and the Houghton groups

TL;DR

This paper proves that all Houghton groups and their finite index subgroups are invariably generated, addressing questions about their subgroup structures and extending understanding of invariable generation in infinite groups.

Contribution

It establishes that each Houghton group and all of its finite index subgroups are invariably generated, filling gaps in the understanding of subgroup properties related to invariable generation.

Findings

All Houghton groups are IG.

Finite index subgroups of Houghton groups are IG.

Addresses questions about subgroup IG properties in infinite groups.

Abstract

The Houghton groups $H_1, H_2, \ldots$ are a family of infinite groups. In 1975 Wiegold showed that $H_3$ was invariably generated (IG) but $H_1\le H_3$ was not. A natural question is then whether the groups $H_2, H_3, \ldots$ are all IG. Wiegold also ends by saying that, in the examples he had found of an IG group with a subgroup that is not IG, the subgroup was never of finite index. Another natural question is then whether there is a subgroup of finite index in $H_3$ that is not IG. In this note we prove, for each $n\in \{2, 3, \ldots\}$, that $H_n$ and all of its finite index subgroups are IG. The independent work of Minasyan and Goffer-Lazarovich in June 2020 frames this note quite nicely: they showed that an IG group can have a finite index subgroup that is not IG.