Non-split linear sharply $2$-transitive groups

TL;DR

This paper constructs examples of countable linear groups in higher dimensions that act sharply 2-transitively without nontrivial normal abelian subgroups, expanding understanding of such groups in linear algebraic settings.

Contribution

It provides the first known examples of linear sharply 2-transitive groups in dimensions three and higher with specific properties, previously only known in non-linear cases.

Findings

Examples of countable linear groups in SL_n(R) for n ≥ 3 with no nontrivial normal abelian subgroups.

These groups admit faithful sharply 2-transitive actions.

The groups have permutational characteristic 2, with involutions acting without fixed points.

Abstract

We give examples of countable linear groups in $SL_{n}(R)$ for $n \ge 3$, 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. Our examples are of permutational characteristic $2$, in the sense that involutions do not fix a point in the $2$-transitive action.