An integral weight realization theorem for subset currents on free groups

We prove that if N ≥ 2 and α : F_N → φ1(γ) is a marking on F_N, then, for any integer r ≥ 2 and any F_N-invariant collection of non-negative integral "weights" associated to all sub-trees K of γ of radius ≤ r satisfying some natural "switch" con-ditions, there exists a finite cyclically reduced folded γ-graph δ realizing these weights as numbers of "occurrences" of K in δ. As an application, we give a new, direct, and explicit proof of one of the main results of our paper with Tatiana Nagnibeda (Subset cur-rents on free groups, Geom. Dedicata 166 (2013), 307-348) stating that, for any N ≥ 2, the set SCurrr(F_N) of all rational subset cur-rents is dense in the space SCurr(F_N) of subset currents on F_N. (The proof given in the above-cited paper was indirect and omitted significant details. The proof given here is complete and, we hope, more accessible to the Out(F_N) community.) We also answer one of the questions (Problem 10.11) posed in the above-mentioned paper. Thus, we prove that if a nonzero μ € SCurr(F_N) has all weights with respect to some marking being integers, then μ is the sum of finitely many "counting" currents corresponding to nontrivial finitely generated subgroups of F_N.