Strictly ascending HNN extensions of finite rank free groups that are linear over Z

TL;DR

This paper constructs specific strictly ascending HNN extensions of finite rank free groups with non-positively curved square complex presentations, proving their hyperbolicity and linearity over the integers, advancing understanding of free group extensions.

Contribution

It introduces new examples of ascending HNN extensions of free groups with non-positively curved complexes, demonstrating their hyperbolicity and linearity over Z.

Findings

Groups are word hyperbolic.

Groups are linear over the integers.

Explicit example of endomorphism provided.

Abstract

We find strictly ascending HNN extensions of finite rank free groups possessing a presentation 2-complex which is a non positively curved square complex. On showing these groups are word hyperbolic, we have by results of Wise and Agol that they are linear over the integers. An example is the endomorphism of the free group on a,b with inverses A,B that sends a to aBaab and b to bAbba.