On the finite index subgroups of Houghton's groups

TL;DR

This paper characterizes all finite index subgroups of Houghton's groups, detailing their isomorphism types and the minimal number of generators needed, with new results and an erratum addressing previous issues.

Contribution

It provides a complete description of finite index subgroups of Houghton groups and characterizes their minimal generating sets, including new results and corrections.

Findings

All finite index subgroups of Houghton groups are described.

The minimal number of generators for these subgroups is either equal to or one more than that of the original group.

Conditions are provided for when each case occurs.

Abstract

An erratum has been added to resolve an issue raised by Professor Derek Holt. This appears after the original paper, and also includes two new results. Original abstract: Houghton's groups $H_2, H_3, \ldots$ are certain infinite permutation groups acting on a countably infinite set; they have been studied, among other things, for their finiteness properties. In this note we describe all of the finite index subgroups of each Houghton group, and their isomorphism types. Using the standard notation that $d(G)$ denotes the minimal size of a generating set for $G$ we then show, for each $n\in \{2, 3,\ldots\}$ and $U$ of finite index in $H_n$, that $d(U)\in\{d(H_n), d(H_n)+1\}$ and characterise when each of these cases occurs.