Extensions of Abelian Automata Groups

A theorem of Nekrashevych and Sidki shows the Mealy Automata structures one can place on Z^m are parametrized by a family of matrices (called "1/2-integral") and a choice of residuation vector e in Z^m. While the impact of the chosen matrix is well understood, the impact of the residuation vector on the resulting structure is seemingly sporadic. In this paper we characterize the impact of the residuation vector e by recognizing an initial structure when e is the first standard basis vector. All other choices of e extend this initial structure by adding "fractional elements" in a way we make precise.