Invariable generation and the Houghton groups

2020/07/03 by Cox, Charles Garnet
#FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2007.01626

Abstract

The Houghton groups H₁, H₂, … are a family of infinite groups. In 1975 Wiegold showed that H₃ was invariably generated (IG) but H₁≤ H₃ was not. A natural question is then whether the groups H₂, H₃, … 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 H₃ that is not IG. In this note we prove, for each n∈ \2, 3, …\, 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.

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