Non-split linear sharply 2-transitive groups

Glasner, Yair; Gulko, Dennis

doi:10.1090/proc/15360

Non-split linear sharply 2-transitive groups
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by Yair Glasner and Dennis D. Gulko

Proc. Amer. Math. Soc. 149 (2021), 2305-2317

DOI: https://doi.org/10.1090/proc/15360

Published electronically: March 16, 2021

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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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