Infinite 32‐generated groups. Issue 4 (10th June 2020)

• Record Type: Journal Article
• Authors: Donoven, Casey; Harper, Scott
• Is Part Of: Bulletin of the London Mathematical Society. Volume 52:Issue 4(2020)
• Page Start: 657
• Page End: 673
• Publication Date: 2020-06-10
• Subjects: 20F05 (primary) -- 20B22 -- 20E32 (secondary); Mathematics -- Periodicals; 510
• DOI: 10.1112/blms.12356
• Languages: English
• ISSNs: 0024-6093

Abstract

Abstract: 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 V n and the Brin–Thompson groups m V are two families of finitely presented groups that generalise V . In this paper, we prove that all of the groups V n, V n ′ and m V 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.