Domains of proper discontinuity on the boundary of outer space. (English) Zbl 1259.20050
The authors construct domains of discontinuity in the compactified Outer space and in the projectivized space of geodesic currents for any “dynamically large” subgroup of \(\mathrm{Out}(F_N)\). The work is motivated by the work of J. McCarthy and A. Papadopoulos [Comment. Math. Helv. 64, No. 1, 133-166 (1989; Zbl 0681.57002)]. As a corollary, the authors prove that for any \(N>3\), the action of \(\mathrm{Out}(F_N)\) on the subset of the projectivized space of geodesic currents consisting of all projectivized currents with full support is properly discontinuous.
Reviewer: Athanase Papadopoulos (Strasbourg)
MSC:
| 20F65 | Geometric group theory |
| 20E36 | Automorphisms of infinite groups |
| 37C85 | Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\) |
| 57M07 | Topological methods in group theory |
Keywords:
outer space; domains of discontinuity; projectivized space of geodesic currents; free group automorphismsCitations:
Zbl 0681.57002References:
| [1] | M. Bestvina and M. Feighn, The topology at infinity of \(\mathrm{Out}(F_n)\) , Invent. Math. 140 (2000), 651-692. · Zbl 0954.55011 · doi:10.1007/s002220000068 |
| [2] | M. Bestvina and M. Feighn, Outer limits , preprint, 1993; available at http://andromeda.rutgers.edu/ feighn/papers/outer.pdf. |
| [3] | M. Bestvina and M. Handel, Train tracks and automorphisms of free groups , Ann. of Math. (2) 135 (1992), 1-51. · Zbl 0757.57004 · doi:10.2307/2946562 |
| [4] | M. Bestvina and M. Feighn, A hyperbolic \(\operatorname{Out}(F_n)\)-complex , Groups Geom. Dyn. 4 (2010), 31-58. · Zbl 1190.20017 · doi:10.4171/GGD/74 |
| [5] | F. Bonahon, Bouts des variétés hyperboliques de dimension 3 , Ann. of Math. (2) 124 (1986), 71-158. JSTOR: · Zbl 0671.57008 · doi:10.2307/1971388 |
| [6] | F. Bonahon, The geometry of Teichmüller space via geodesic currents , Invent. Math. 92 (1988), 139-162. · Zbl 0653.32022 · doi:10.1007/BF01393996 |
| [7] | F. Bonahon, Geodesic currents on negatively curved groups , Arboreal group theory (Berkeley, CA, 1988), Math. Sci. Res. Inst. Publ., vol. 19, Springer, New York, 1991, pp. 143-168. · Zbl 0772.57004 |
| [8] | P. Brinkmann, Hyperbolic automorphisms of free groups , Geom. Funct. Anal. 10 (2000), 1071-1089. · Zbl 0970.20018 · doi:10.1007/PL00001647 |
| [9] | M. Cohen and M. Lustig, Very small group actions on \(R\)-trees and Dehn twist automorphisms , Topology 34 (1995), 575-617. · Zbl 0844.20018 · doi:10.1016/0040-9383(94)00038-M |
| [10] | T. Coulbois, A. Hilion and M. Lustig, Non-unique ergodicity, observers’ topology and the dual algebraic lamination for \(\mathbb R\)-trees , Illinois J. Math. 51 (2007), 897-911. · Zbl 1197.20020 |
| [11] | T. Coulbois, A. Hilion and M. Lustig, \(\mathbb R\)-trees and laminations for free groups II: The dual lamination of an \(\mathbb R\)-tree , J. Lond. Math. Soc. (2) 78 (2008), 737-754. · Zbl 1198.20023 · doi:10.1112/jlms/jdn053 |
| [12] | T. Coulbois, A. Hilion and M. Lustig, \(\mathbb R\)-trees and laminations for free groups III: Currents and dual \(\mathbb R\)-tree metrics , J. Lond. Math. Soc. (2) 78 (2008), 755-766. · Zbl 1200.20018 · doi:10.1112/jlms/jdn054 |
| [13] | M. Culler and K. Vogtmann, Moduli of graphs and automorphisms of free groups , Invent. Math. 84 (1986), 91-119. · Zbl 0589.20022 · doi:10.1007/BF01388734 |
| [14] | B. Farb and L. Mosher, Convex cocompact subgroups of mapping class groups , Geom. Topol. 6 (2002), 91-152. · Zbl 1021.20034 · doi:10.2140/gt.2002.6.91 |
| [15] | S. Francaviglia, Geodesic currents and length compactness for automorphisms of free groups , Trans. Amer. Math. Soc. 361 (2009), 161-176. · Zbl 1166.20032 · doi:10.1090/S0002-9947-08-04420-6 |
| [16] | V. Guirardel, Approximations of stable actions on \(R\)-trees , Comment. Math. Helv. 73 (1998), 89-121. · Zbl 0979.20026 · doi:10.1007/s000140050047 |
| [17] | V. Guirardel, Dynamics of \(\operatorname{Out}(F_ n)\) on the boundary of outer space , Ann. Sci. École Norm. Sup. (4) 33 (2000), 433-465. · Zbl 1045.20034 · doi:10.1016/S0012-9593(00)00117-8 |
| [18] | U. Hamenstädt, Word hyperbolic extensions of surface groups , preprint, 2005; available at |
| [19] | U. Hamenstädt, Lines of minima in Outer space , preprint, 2009; available at · Zbl 1209.37023 · doi:10.1007/s00222-008-0163-5 |
| [20] | I. Kapovich, The frequency space of a free group , Internat. J. Algebra Comput. 15 (2005), 939-969. · Zbl 1110.20031 · doi:10.1142/S0218196705002700 |
| [21] | I. Kapovich, Currents on free groups , Topological and asymptotic aspects of group theory (R. Grigorchuk, M. Mihalik, M. Sapir and Z. Sunik, eds.), Contemp. Math., vol. 394, Amer. Math. Soc., Providence, RI, 2006, pp. 149-176. · Zbl 1110.20034 |
| [22] | I. Kapovich, Clusters, currents and Whitehead’s algorithm , Experiment. Math. 16 (2007), 67-76. · Zbl 1158.20014 · doi:10.1080/10586458.2007.10128990 |
| [23] | I. Kapovich and M. Lustig, The actions of \(\operatorname{Out}(F_k)\) on the boundary of Outer space and on the space of currents: Minimal sets and equivariant incompatibility , Ergodic Theory Dynam. Systems 27 (2007), 827-847. · Zbl 1127.20025 · doi:10.1017/S0143385706001015 |
| [24] | I. Kapovich and M. Lustig, Geometric Intersection Number and analogues of the Curve Complex for free groups , Geom. Topol. 13 (2009), 1805-1833. · Zbl 1194.20046 · doi:10.2140/gt.2009.13.1805 |
| [25] | I. Kapovich and M. Lustig, Intersection form, laminations and currents on free groups , Geom. Funct. Anal. 19 (2010), 1426-1467. · Zbl 1242.20052 · doi:10.1007/s00039-009-0041-3 |
| [26] | I. Kapovich and M. Lustig, Ping-pong and Outer space , Journal of Topology and Analysis 2 (2010), 173-201. · Zbl 1211.20027 · doi:10.1142/S1793525310000318 |
| [27] | I. Kapovich and M. Lustig, Stabilizers of \(\mathbb R\)-trees with free isometric actions of \(F_N\) , to appear in Journal of Group Theory; available at arXiv :0904.1881. · Zbl 1262.20031 |
| [28] | I. Kapovich and T. Nagnibeda, The Patterson-Sullivan embedding and minimal volume entropy for Outer space , Geom. Funct. Anal. 17 (2007), 1201-1236. · Zbl 1135.20031 · doi:10.1007/s00039-007-0621-z |
| [29] | R. P. Kent and C. J. Leininger, Subgroups of mapping class groups from the geometrical viewpoint , In the tradition of Ahlfors-Bers. IV, Contemp. Math., vol. 432, Amer. Math. Soc., Providence, RI, 2007, pp. 119-141. · Zbl 1140.30017 |
| [30] | R. P. Kent and C. J. Leininger, Shadows of mapping class groups: Capturing convex cocompactness , Geom. Funct. Anal. 18 (2008), 1270-1325. · Zbl 1282.20046 · doi:10.1007/s00039-008-0680-9 |
| [31] | R. P. Kent and C. J. Leininger, Uniform convergence in the mapping class group , Ergodic Theory Dynam. Systems 28 (2008), 1177-1195. · Zbl 1153.57013 · doi:10.1017/S0143385707000818 |
| [32] | G. Levitt and M. Lustig, Irreducible automorphisms of \(F_n\) have North-South dynamics on compactified Outer space , J. Inst. Math. Jussieu 2 (2003), 59-72. · Zbl 1034.20038 · doi:10.1017/S1474748003000033 |
| [33] | R. Martin, Non-uniquely ergodic foliations of thin type, measured currents and automorphisms of free groups , Ph.D. Thesis, University of Utah, 1995. |
| [34] | H. Masur, Measured foliations and handlebodies , Ergodic Theory Dynam. Systems 6 (1986), 99-116. · Zbl 0628.57010 · doi:10.1017/S014338570000331X |
| [35] | J. McCarthy and A. Papadopoulos, Dynamics on Thurston’s sphere of projective measured foliations , Comment. Math. Helv. 64 (1989), 133-166. · Zbl 0681.57002 · doi:10.1007/BF02564666 |
| [36] | F. Paulin, The Gromov topology on \(R\)-trees , Topology Appl. 32 (1989), 197-221. · Zbl 0675.20033 · doi:10.1016/0166-8641(89)90029-1 |
| [37] | K. Vogtmann, Automorphisms of free groups and Outer space , Geom. Dedicata 94 (2002), 1-31. · Zbl 1017.20035 · doi:10.1023/A:1020973910646 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
