The finiteness conjecture holds in SL(2,Z>=0)^2

TL;DR

This paper proves the Lagarias-Wang finiteness conjecture for certain pairs of matrices in SL(2,Z>=0), establishing conditions under which the conjecture holds and identifying optimal products.

Contribution

It demonstrates the conjecture's validity for matrix pairs in SL(2,Z>=0) under specific orientation and entry conditions, extending previous results.

Findings

Finiteness conjecture holds for pairs in SL(2,Z>=0) under given conditions.

Optimal products identified as {A,B,AB,A^2B,AB^2}.

Conditions include coherent orientation and integer entries.

Abstract

Let A,B be matrices in SL(2,R) having trace greater than or equal to 2. Assume the pair A,B is coherently oriented, that is, can be conjugated to a pair having nonnegative entries. Assume also that either A,B^(-1) is coherently oriented as well, or A,B have integer entries. Then the Lagarias-Wang finiteness conjecture holds for the set {A,B}, with optimal product in {A,B,AB,A^2B,AB^2}. In particular, it holds for every matrix pair in SL(2,Z>=0).