On the Density of Eigenvalues on Periodic Graphs

Suppose that is a graph with vertices , edges free group action on the finitely many orbits, and linear operator of finite order Hilbert space such commutes action. Fix in pure-point spectrum consider vector all eigenfunctions support . Then nontrivial generated module over ring Laurent polynomials. We show density given by an Euler-characteristic-type formula taking resolution Furthermore, we generalize these claims under suitable assumptions to noncommutative setting amenable acting freely orbits commuting Finally, provide example, abelian case, where nontrivial.