Current twisting and nonsingular matrices

TL;DR

This paper demonstrates that for free groups of rank at least 3, any invertible integer matrix can be realized as the action of a hyperbolic fully irreducible automorphism, linking algebraic and dynamical properties.

Contribution

It establishes the existence of hyperbolic fully irreducible automorphisms with prescribed actions on Z^k for matrices in GL(k,Z), for k ≥ 3.

Findings

Any matrix in GL(k,Z) can be realized as an automorphism's induced action for k ≥ 3.

Constructs hyperbolic fully irreducible automorphisms corresponding to given matrices.

Links algebraic matrices with dynamical automorphisms in free groups.

Abstract

We show that for k at least 3, given any matrix in GL(k,Z), there is a hyperbolic fully irreducible automorphism of the free group of rank k whose induced action on Z^k is the given matrix.