Every ﬁnite simple group can be generated by two elements, and Guralnick and Kantor proved that, moreover, every nontrivial element is contained in a generating pair. Groups with this property are said to be 3 2-generated. Thompson’s group V was the ﬁrst ﬁnitely presented inﬁnite simple group to be discovered. The Higman–Thompson groups Vn and the Brin–Thompson groups mV are two families of ﬁnitely presented groups that generalise V. In this paper, we prove that all of the groups Vn, V0 n and mV are 3 2-generated. As far as the authors are aware, the only previously known examples of inﬁnite noncyclic 3 2-generated groups are the pathological Tarski monsters. We conclude with several open questions motivated by our results.
- 20F05 (primary)
- 20E32 (secondary)