TY - JOUR

T1 - An integral weight realization theorem for subset currents on free groups

AU - Kapovich, Ilya

N1 - Funding Information:
The author was supported by the NSF grant DMS-1405146.
Publisher Copyright:
© 2016 Topology Proceedings.

PY - 2017

Y1 - 2017

N2 - We prove that if N ≥ 2 and α : FN → φ1(γ) is a marking on FN, then, for any integer r ≥ 2 and any FN-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(FN) of all rational subset cur-rents is dense in the space SCurr(FN) of subset currents on FN. (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(FN) 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(FN) 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 FN.

AB - We prove that if N ≥ 2 and α : FN → φ1(γ) is a marking on FN, then, for any integer r ≥ 2 and any FN-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(FN) of all rational subset cur-rents is dense in the space SCurr(FN) of subset currents on FN. (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(FN) 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(FN) 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 FN.

KW - Free groups

KW - Geodesic currents

KW - Rauzy-de Bruijn graphs.

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M3 - Article

AN - SCOPUS:85018609406

VL - 50

SP - 213

EP - 236

JO - Topology Proceedings

JF - Topology Proceedings

SN - 0146-4124

ER -