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Subset currents on free groups. (English) Zbl 1288.20053
Let \(F_N\) be the free group on \(N\) generators. The geodesic currents on \(F_N\), denoted \(\text{Curr}(F_N)\), is the set of all positive, \(F_N\)-invariant, locally finite Borel measures defined on the set of two-element subsets of \(\partial F_N\). This set was used in the study of properties of the outer space of \(F_N\). The elements of \(\text{Curr}(F_N)\) are the analogues of the conjugacy classes of elements in \(F_N\). In the paper under review, the authors introduce a generalization of \(\text{Curr}(F_N)\), the subset currents, denoted by \(\mathcal S\text{Curr}(F_N)\), to be the set of all positive, \(F_N\)-invariant, locally finite Borel measures on the set \(\mathfrak C_N\) of all subset of \(\partial F_N\) containing at least two elements. The authors generalize the constructions on \(\text{Curr}(F_N)\) to \(\mathcal S\text{Curr}(F_N)\) and show that \(\mathcal S\text{Curr}(F_N)\) has much richer properties. As a first remark, the elements of \(\mathcal S\text{Curr}(F_N)\) are the analogues of conjugacy classes of finitely generated subgroups of \(F_N\). The space \(\mathcal S\text{Curr}(F_N)\) admits the weak \(*\)-topology of convergence of integrals of continuous functions with compact support.
The space \(\mathcal S\text{Curr}(F_N)\) admits an action of \(\text{Out}(F_N)\). For \(\mu\in\mathcal S\text{Curr}(F_N)\), \(U\in\mathfrak C_N\), and \(\varphi\in\text{Out}(F_N)\), \(\varphi\mu(U)=\mu(\varphi^{-1}(U))\). That makes \(\text{Curr}(F_N)\) a closed \(\text{Out}(F_N)\)-invariant subset.
Let \(H\) be a finitely generated nontrivial subgroup of \(F_N\), the authors define the counting measure of \(H\). Let \(\text{Comm}_{F_N}(H)\) be the virtual normalizer of \(H\). If \(H=\text{Comm}_{F_N}(H)\), define \(\eta_H\in\mathcal S\text{Curr}(F_N)\) to be the sum, over the conjugates of \(H\), of the \(\delta\) measures of the limit sets of the conjugates. In general, if \(H_0\) is the virtual normalizer of \(H\) and \(H\) has index \(m\) in \(H_0\), then \(\eta_H=m\eta_{H_0}\). Positive multiples of the measures \(\eta_H\) are called rational subset currents. The set of rational subset currents is a dense subset of \(\mathcal S\text{Curr}(F_N)\), generalizing the corresponding result for geodesic currents. Using \(\eta_H\), the authors define a pairing on \(\text{cv}_N\times\mathcal S\text{Curr}(F_N)\) with values in \(\mathbb R_{\geq 0}\), where \(\text{cv}_N\) is the outer space of \(F_N\). If \(T\) is a tree, representing an element of \(\text{cv}_N\) and \(\eta_H\) a counting measure, then \(\langle T,\eta_H\rangle=\text{vol}(T_H/H)\), where \(T_H\) is a minimal \(H\)-invariant subtree of \(T\). This covolume pairing generalizes the corresponding pairing on \(\text{Curr}(F_N)\). The only difference is that the new pairing does not extend to the compactification of \(\text{cv}_N\), as it does the old one.
For a finitely generated subgroup \(F\) of \(F_N\), define the reduced rank \(\overline{\text{rk}}(F)\) to be \(\text{rk}F-1\), if \(F\) is not trivial and \(0\) if it is trivial. Then the function \(\overline{\text{rk}}\) defined on \(\mathcal S\text{Curr}(F_N)\) by the rule \(\overline{\text{rk}}(\eta_H)=\overline{\text{rk}}(H)\) is an \(\mathbb R_{\geq 0}\)-linear, \(\text{Out}(F_N)\)-invariant functional.
Finally, the authors ask a series of questions that lead to further investigating the properties of the subset currents, as for example the extension of the constructions to general word hyperbolic groups.

MSC:
20F65 Geometric group theory
20E05 Free nonabelian groups
20F28 Automorphism groups of groups
20E08 Groups acting on trees
57M07 Topological methods in group theory
37A35 Entropy and other invariants, isomorphism, classification in ergodic theory
37B10 Symbolic dynamics
37E25 Dynamical systems involving maps of trees and graphs
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References:
[1] Abert, M., Glasner, Y., Virag, B.: Kesten’s theorem for invariant random subgroups, preprint. arXiv:1201.3399 (2012) · Zbl 1344.20061
[2] Arnoux, P.; Berthé, V.; Fernique, T.; Jamet, D., Functional stepped surfaces, flips, and generalized substitutions, Theor. Comput. Sci., 380, 251-265, (2007) · Zbl 1119.68136
[3] Bassino, F.; Nicaud, C.; Weil, P.:, Random generation of finitely generated subgroups of a free group, Int. J. Algebra Comput., 18, 375-405, (2008) · Zbl 1193.05017
[4] Benjamini, I.; Schramm, O., Recurrence of distributional limits of finite planar graphs, Electron. J. Probab., 6, 1-23, (2001) · Zbl 1010.82021
[5] Bestvina, M., Feighn, M.: Outer Limits, preprint. http://andromeda.rutgers.edu/ feighn/papers/outer.pdf (1993) · Zbl 1125.20020
[6] Bestvina, M.; Feighn, M.; Handel, M., Laminations, trees, and irreducible automorphisms of free groups, Geom. Funct. Anal., 7, 215-244, (1997) · Zbl 0884.57002
[7] Bestvina, M.; Feighn, M., A hyperbolic out(\(F\)_{\(n\)}) complex, Groups Geom. Dyn., 4, 31-58, (2010) · Zbl 1190.20017
[8] Bonahon, F., Bouts des variétés hyperboliques de dimension 3, Ann. of Math.(2), 124, 71-158, (1986) · Zbl 0671.57008
[9] Bonahon, F., The geometry of Teichmüller space via geodesic currents, Invent. Math., 92, 139-162, (1988) · Zbl 0653.32022
[10] Bowen, L., Periodicity and circle packings of the hyperbolic plane, Geom. Dedicata, 102, 213-236, (2003) · Zbl 1074.52007
[11] Bowen, L., Free groups in lattices, Geom. Topol., 13, 3021-3054, (2009) · Zbl 1244.22003
[12] Bowen, L.: Random walks on coset spaces with applications to Furstenberg entropy, preprint. arXiv:1008.4933 (2010) · Zbl 1239.20031
[13] Bowen, L.: Invariant random subgroups of the free group, preprint. arXiv:1204.5939 (2012) · Zbl 1242.20052
[14] Carette, M., Francaviglia, S., Kapovich, I., Martino, A.: Spectral rigidity of automorphic orbits in free groups. Alg. Geom. Topol. 12, 1457-1486 (2012) · Zbl 1261.20040
[15] Cohen, M.; Lustig, M., Very small group actions on \(R\)-trees and Dehn twist automorphisms, Topology, 34, 575-617, (1995) · Zbl 0844.20018
[16] Clay, M.; Pettet, A., Currents twisting and nonsingular matrices, Commentarii Mathematici Helvetici, 87, 384-407, (2012) · Zbl 1286.20049
[17] Coulbois, T.; Hilion, A.; Lustig, M., \({{\mathbb{R}}}\)-trees and laminations for free groups I: algebraic laminations, J. Lond. Math. Soc. (2), 78, 723-736, (2008) · Zbl 1197.20019
[18] Coulbois, T.; Hilion, A.; Lustig, M., \({{\mathbb{R}}}\)-trees and laminations for free groups II: the dual lamination of an \({{\mathbb{R}}}\)-tree, J. Lond. Math. Soc. (2), 78, 737-754, (2008) · Zbl 1198.20023
[19] Coulbois, T.; Hilion, A.; Lustig, M., \({{\mathbb{R}}}\)-trees and laminations for free groups III: currents and dual \({{\mathbb{R}}}\)-tree metrics, J. Lond. Math. Soc. (2), 78, 755-766, (2008) · Zbl 1200.20018
[20] Culler, M.; Vogtmann, K., Moduli of graphs and automorphisms of free groups, Invent. Math., 84, 91-119, (1986) · Zbl 0589.20022
[21] D’Angeli, D.; Donno, A.; Matter, M.; Nagnibeda, T., Schreier graphs of the basilica group, J. Mod. Dyn., 4, 167-205, (2010) · Zbl 1239.20031
[22] Dani, S.G., On conjugacy classes of closed subgroups and stabilizers of Borel actions of Lie groups, Ergod. Theory Dyn. Syst., 22, 1697-1714, (2002) · Zbl 1016.22013
[23] Elek, G., On the limit of large girth graph sequences, Combinatorica, 30, 553-563, (2010) · Zbl 1231.05259
[24] Francaviglia, S., Geodesic currents and length compactness for automorphisms of free groups, Trans. Am. Math. Soc., 361, 161-176, (2009) · Zbl 1166.20032
[25] Grigorchuk, R., Some topics of dynamics of group actions on rooted trees, Proc. Steklov Inst. Math., 273, 64-175, (2011) · Zbl 1268.20027
[26] Grigorchuk, R.; Kaimanovich, V.A.; Nagnibeda, T., Ergodic properties of boundary actions and Nielsen-Schreier theory, Adv. Math., 230, 1340-1380, (2012) · Zbl 1278.37034
[27] Guirardel, V., Dynamics of out (\(F\)_{\(n\)}) on the boundary of outer space, Ann. Sci. École Norm. Sup. (4), 33, 433-465, (2000) · Zbl 1045.20034
[28] Hamenstädt, U.: Lines of minima in outer space. November 2009, preprint. arXiv:0911.3620 (2009) · Zbl 1286.20049
[29] Hart K.P., Nagata J.-I., Vaughan J.E. (ed.): Encyclopedia of General Topology. Elsevier, Amsterdam (2004) · Zbl 1059.54001
[30] Kapovich, I.: Currents on free groups. In: Grigorchuk, R., Mihalik, M., Sapir, M., Sunik, Z. (eds.) Topological and Asymptotic Aspects of Group Theory, AMS Contemporary Mathematics Series, vol. 394, pp. 149-176 (2006) · Zbl 1110.20034
[31] Kapovich, I., Clusters, currents and whitehead’s algorithm, Exp. Math., 16, 67-76, (2007) · Zbl 1158.20014
[32] Kapovich, I., Random length-spectrum rigidity for free groups, Proc. AMS, 140, 1549-1560, (2012) · Zbl 1268.20043
[33] Kapovich, I.; Levitt, G.; Schupp, P.; Shpilrain, V., Translation equivalence in free groups, Trans. Am. Math. Soc., 359, 1527-1546, (2007) · Zbl 1119.20037
[34] Kapovich, I.; Lustig, M., The actions of out(\(F\)_{\(k\)}) on the boundary of outer space and on the space of currents: minimal sets and equivariant incompatibility, Ergod. Theory Dyn. Syst., 27, 827-847, (2007) · Zbl 1127.20025
[35] Kapovich, I.; Lustig, M., Geometric intersection number and analogues of the curve complex for free groups, Geom. Topol., 13, 1805-1833, (2009) · Zbl 1194.20046
[36] Kapovich, I.; Lustig, M., Intersection form, laminations and currents on free groups, Geom. Funct. Anal. (GAFA), 19, 1426-1467, (2010) · Zbl 1242.20052
[37] Kapovich, I., Lustig, M.: Domains of proper dicontinuity on the boundary of outer space. Ill. J. Math. 54(1):89-108. Special issue dedicated to Paul Schupp (2010) · Zbl 1259.20050
[38] Kapovich, I.; Lustig, M., Ping-pong and outer space, J. Topol. Anal., 2, 173-201, (2010) · Zbl 1211.20027
[39] Kapovich, I.; Myasnikov, A., Stallings foldings and the subgroup structure of free groups, J. Algebra, 248, 608-668, (2002) · Zbl 1001.20015
[40] Kapovich, I.; Nagnibeda, T., The patterson-Sullivan embedding and minimal volume entropy for outer space, Geom. Funct. Anal. (GAFA), 17, 1201-1236, (2007) · Zbl 1135.20031
[41] Kapovich, I., Nagnibeda, T.: Geometric Entropy of Geodesic Currents on Free Groups. Dynamical Numbers: Interplay Between Dynamical Systems and Number Theory, Contemporary Mathematics Series. American Mathematical Society, Providence, RI, pp. 149-176 (2010) · Zbl 1216.20034
[42] Kapovich, I.; Short, H., Greenberg’s theorem for quasiconvex subgroups of word hyperbolic groups, Can. J. Math., 48, 1224-1244, (1996) · Zbl 0873.20025
[43] Lee, D., Translation equivalent elements in free groups, J. Group Theory, 9, 809-814, (2006) · Zbl 1112.20023
[44] Lee, D., An algorithm that decides translation equivalence in a free group of rank two, J. Group Theory, 10, 561-569, (2007) · Zbl 1125.20020
[45] Lee, D.; Ventura, E., Volume equivalence of subgroups of free groups, J. Algebra, 324, 195-217, (2010) · Zbl 1207.20014
[46] Levitt, G.; Lustig, M., Irreducible automorphisms of \(F\)_{\(n\)} have north-south dynamics on compactified outer space, J. Inst. Math. Jussieu, 2, 59-72, (2003) · Zbl 1034.20038
[47] Martin, R.: Non-uniquely ergodic foliations of thin type, measured currents and automorphisms of free groups. Ph.D. Thesis (1995) · Zbl 0836.22018
[48] Mineyev, I., Submultiplicativity and the hanna Neumann conjecture, Ann. Math. (2), 175, 393-414, (2012) · Zbl 1280.20029
[49] Savchuk, D.: Schreier Graphs of actions of Thompsons Group \(F\) on the unit interval and on the cantor set, preprint. arXiv:1105.4017 (2011) · Zbl 1280.20029
[50] Stallings, J., Topology of finite graphs, Invent. Math., 71, 551-565, (1983) · Zbl 0521.20013
[51] Stuck, G.; Zimmer, R.J., Stabilizers for ergodic actions of higher rank semisimple groups, Ann. Math. (2), 139, 723-747, (1994) · Zbl 0836.22018
[52] Vershik, A.: Nonfree actions of countable groups and their characters. arXiv:1012.4604 · Zbl 1279.37004
[53] Vershik, A.: Totally nonfree actions and infinite symmetric group, preprint. arXiv:1109.3413 (2011) · Zbl 1119.68136
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