Phylogenetic invariants for group-based models

In this paper we investigate properties of algebraic varieties representing group-based phylogenetic models. We propose a method of generating many phylogenetic invariants. We prove that we obtain all invariants for any tree for the binary Jukes-Cantor model. We conjecture that our method can give all phylogenetic invariants for any tree. We show that for 3-Kimura our conjecture is equivalent to the conjecture of Sturmfels and Sullivant. This, combined with the results of Sturmfels and Sullivant, would make it possible to determine all phylogenetic invariants for any tree for 3-Kimura model, and also other phylogenetic models. Next we give the (first) example of a non-normal general group-based model for an abelian group. Following Kubjas we also determine some invariants of group-based models showing that the associated varieties do not have to be deformation equivalent.