Finite phylogenetic complexity of ZpZp and invariants for Z3Z3

We study phylogenetic complexity of finite abelian groups—an invariant introduced by Sturmfels and Sullivant (2005). The invariant is hard to compute—so far it was only known for Z2Z2, in which case it equals 22 (Sturmfels and Sullivant, 2005), (Chifman and Petrović, 2007). We prove that phylogenetic complexity of any group ZpZp, where pp is prime, is finite. We also show, as conjectured by Sturmfels and Sullivant, that the phylogenetic complexity of Z3Z3 equals 33.