The finiteness conjecture holds in SL(2,Z>=0)^2 (2006.16899v2)

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).