On the finite index subgroups of Houghton\u2019s groups

Charles Garnet Cox

doi:10.1007/s00013-021-01677-y

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 nin {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.

Highlights

Introduced in [11] by Houghton, the Houghton groups have since attracted attention for their finiteness properties [2,13], their growth [3,10], their many interesting combinatorial features [1,9,14,15] as well as other properties.Definition
We give a brief overview of these groups for our purposes; more detailed introductions can be found, for example, in [1,4]
We extend the work in [7], where the groups Alt(X2), g2c were shown to be 2-generated for each c ∈ N, by investigating the generation properties of each of these groups

Summary

Introduction

Introduced in [11] by Houghton, the Houghton groups have since attracted attention for their finiteness properties [2,13], their growth [3,10], their many interesting combinatorial features [1,9,14,15] as well as other properties.Definition. For a finitely generated group G, let d(G) := min{|S| : S = G}. Our second theorem states that there is a close connection between the minimal number of generators of a Houghton group and its finite index subgroups. This complete categorisation provides us with subgroups of the Houghton groups with constant minimal number of generators on finite index subgroups.

Results

Conclusion