On the linearity of the holomorph group of a free group on two generators

TL;DR

This paper proves that the holomorph of the free group on two generators is linear, leading to new linearity results for related automorphism groups and mapping class groups.

Contribution

It establishes the linearity of Hol(F_2), enabling linearity of certain split extensions and mapping class groups, a novel result in group theory.

Findings

Hol(F_2) is linear.

Split extensions of F_2 by linear groups are linear.

Mapping class group for genus one surfaces with two punctures is linear.

Abstract

Let F_n denote the free group generated by n letters. The purpose of this article is to show that Hol(F_2), the holomorph of the free group on two generators, is linear. Consequently, any split group extension of F_2 by a linear group H is linear. This result gives a large linear subgroup of Aut(F_3). A second application is that the mapping class group for genus one surfaces with two punctures is linear.