On the spread of infinite groups

Charles Garnet Cox

Published 2020-12-22Version 1

A group is $\frac32$-generated if every non-trivial element is part of a generating pair. In 2019 Donoven and Harper showed that many Thompson groups are $\frac32$-generated and posed five questions. The first of these is whether there exists a 2-generated group with every proper quotient cyclic that is not $\frac32$-generated. This is a natural question given the significant work in proving that no finite group has this property, but we show that there is such an infinite group. The groups we consider are a family of finite index subgroups of the Houghton group FSym$(\mathbb{Z})\rtimes\mathbb{Z}$. We then show that the first two groups in our family are $\frac32$-generated, and investigate the related notion of spread for these groups. We are able to show that they have finite spread which is greater than 2. Other than $\mathbb{Z}$ and the Tarski monsters, neither of these properties have been proven for any infinite group before.