Finitely generated simple sharply 2-transitive groups

TL;DR

This paper constructs the first examples of finitely generated infinite sharply 2-transitive groups, including one with Kazhdan's property (T), simplicity, and minimal conjugacy classes, advancing understanding of such groups' structure.

Contribution

It provides the first known finitely generated infinite sharply 2-transitive groups and constructs a simple example with Kazhdan property (T) and minimal conjugacy classes.

Findings

Existence of finitely generated infinite sharply 2-transitive groups.

Construction of a simple group with Kazhdan property (T).

Group has exactly four conjugacy classes, proven to be minimal.

Abstract

We construct the first examples of infinite sharply 2-transitive groups which are finitely generated. Moreover, we construct such a group that has Kazhdan property (T), is simple, has exactly four conjugacy classes, and we show that this number is as small as possible.