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Non-split linear sharply 2-transitive groups


Authors: Yair Glasner and Dennis D. Gulko
Journal: Proc. Amer. Math. Soc. 149 (2021), 2305-2317
MSC (2020): Primary 20B22
DOI: https://doi.org/10.1090/proc/15360
Published electronically: March 16, 2021
MathSciNet review: 4246784
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Abstract: We give examples of countable linear groups $\Gamma < \operatorname {SL}_{{\mathbf {3}}}({\mathbf {R}})$, with no nontrivial normal abelian subgroups, that admit a faithful sharply $2$-transitive action on a set. Without the linearity assumption, such groups were recently constructed by Rips, Segev, and Tent in [J. Eur. Math. Soc. 19 (2017), pp. 2895–2910]. Our examples are of permutational characteristic $2$, in the sense that involutions do not fix a point in the $2$-transitive action.


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Additional Information

Yair Glasner
Affiliation: Department of Mathematics, Ben-Gurion University of the Negev, P.O.B. 653, Be’er Sheva 84105, Israel
MR Author ID: 673281
ORCID: 0000-0002-6231-3817
Email: yairgl@math.bgu.ac.il

Dennis D. Gulko
Affiliation: Department of Mathematics, Ben-Gurion University of the Negev, P.O.B. 653, Be’er Sheva 84105, Israel
MR Author ID: 1065319
Email: gulkod@math.bgu.ac.il

Keywords: Sharply $2$-transitive, linear groups
Received by editor(s): October 30, 2019
Received by editor(s) in revised form: July 26, 2020
Published electronically: March 16, 2021
Additional Notes: The research of both authors was partially funded by Israel Science Foundation grant ISF 2919/19.
Communicated by: Martin Liebeck
Article copyright: © Copyright 2021 American Mathematical Society