Non-split sharply 2- and 3-transitive groups in SL_n(\mathbb Z)

TL;DR

The paper investigates the existence of non-split sharply 2- and 3-transitive subgroups within special linear groups over integers, providing new examples and non-existence results.

Contribution

It constructs explicit non-split sharply 2- and 3-transitive subgroups in SL_3(Z) and SL_4(Z), and proves non-existence of infinite sharply 3-transitive subgroups in SL_3(Z).

Findings

SL_3(Z) contains a non-split sharply 2-transitive subgroup

SL_4(Z) contains a non-split sharply 3-transitive subgroup

SL_3(Z) does not contain an infinite sharply 3-transitive subgroup

Abstract

We prove that $\mathrm{SL}_3(\mathbb{Z})$ contains a non-split sharply 2-transitive subgroup, answering a question of Glasner and Gulko. We also prove that $\mathrm{SL}_4(\mathbb{Z})$ contains a non-split sharply 3-transitive subgroup, but that $\mathrm{SL}_3(\mathbb{Z})$ does not contain an infinite sharply 3-transitive subgroup.