Tropical matrix groups

We study the structure of groups of finitary tropical matrices under multiplication. We show that the maximal groups of n× n tropical matrices are precisely the groups of the form G× R where G is a group admitting a 2-closed permutation representation on n points. Each such maximal group is also naturally isomorphic to the full linear automorphism group of a related tropical polytope. Our results have numerous corollaries, including the fact that every automorphism of a projective (as a module) tropical polytope of full rank extends to an automorphism of the containing space.