Rank of mapping tori and companion matrices

TL;DR

This paper investigates the algebraic properties of mapping tori derived from matrices in $GL(d,Z)$, providing decidability results for their generation by two elements and classifying generating pairs, with implications for the structure of these groups.

Contribution

It establishes decidability of two-element generation for mapping tori and classifies generating pairs up to Nielsen equivalence, also analyzing the behavior of powers of matrices in $GL(d,Z)$.

Findings

Decidability of two-element generation for certain mapping tori.

Classification of generating pairs up to Nielsen equivalence.

For infinite order matrices, high powers are not conjugate to companion matrices.

Abstract

Given $f$ in $GL(d,Z)$, it is decidable whether its mapping torus (the semi-direct product of $Z^d$ with $Z$) may be generated by two elements or not; if so, one can classify generating pairs up to Nielsen equivalence. If $f$ has infinite order, the mapping torus of $f^n$ cannot be generated by two elements for $n$ large enough; equivalently, $f^n$ is not conjugate to a companion matrix in $GL(d,Z)$ if $n$ is large.