Tropical matrices and group representations

TL;DR

This paper classifies all subgroups of tropical matrix semigroups, proving that each contains a torsion-free abelian subgroup of bounded index, thus resolving a conjecture in the field.

Contribution

It provides a complete description of tropical matrix subgroups and confirms a conjecture about their structure and subgroup properties.

Findings

Every tropical matrix subgroup has a torsion-free abelian subgroup of index at most n!

The classification is complete up to isomorphism

Confirms the Johnson and Kambites conjecture

Abstract

The paper gives a complete description of the subgroups of the semigroup of tropical n-by-n matrices up to an isomorphism. In particular, we show that every of these groups has a torsion-free abelian subgroup of index at most n!, proving the conjecture of Johnson and Kambites.