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Subset currents on surfaces
About this Title
Dounnu Sasaki
Publication: Memoirs of the American Mathematical Society
Publication Year:
2022; Volume 278, Number 1368
ISBNs: 978-1-4704-5343-5 (print); 978-1-4704-7168-2 (online)
DOI: https://doi.org/10.1090/memo/1368
Published electronically: May 23, 2022
Keywords: Subset current,
Geodesic current,
Hyperbolic surface,
Intersection number,
Surface group,
Free group,
Hyperbolic group
Table of Contents
Chapters
- 1. Introduction
- 2. Subset Currents on Hyperbolic Groups
- 3. Volume Functionals on Kleinian Groups
- 4. Subgroups, Inclusion Maps and Finite Index Extension
- 5. Intersection Number
- 6. Intersection Functional on Subset Currents
- 7. Projection from Subset Currents onto Geodesic Currents
- 8. Denseness Property of Rational Subset Currents
Abstract
Subset currents on hyperbolic groups were introduced by Kapovich and Nagnibeda as a generalization of geodesic currents on hyperbolic groups, which were introduced by Bonahon and have been successfully studied in the case of the fundamental group $\pi _1 (\Sigma )$ of a compact hyperbolic surface $\Sigma$. Kapovich and Nagnibeda particularly studied subset currents on free groups. In this article, we develop the theory of subset currents on $\pi _1(\Sigma )$, which we call subset currents on $\Sigma$. We prove that the space $\mathrm {SC}(\Sigma )$ of subset currents on $\Sigma$ is a measure-theoretic completion of the set of conjugacy classes of non-trivial finitely generated subgroups of $\pi _1 (\Sigma )$, each of which geometrically corresponds to a convex core of a covering space of $\Sigma$. This result was proved by Kapovich-Nagnibeda in the case of free groups, and is also a generalization of Bonahon’s result on geodesic currents on hyperbolic groups. We will also generalize several other results of them. Especially, we extend the (geometric) intersection number of two closed geodesics on $\Sigma$ to the intersection number of two convex cores on $\Sigma$ and, in addition, to a continuous $\mathbb {R}_{\geq 0}$-bilinear functional on $\mathrm {SC}(\Sigma )$.- Anja Bankovic and Christopher J. Leininger, Marked-length-spectral rigidity for flat metrics, Trans. Amer. Math. Soc. 370 (2018), no. 3, 1867–1884. MR 3739194, DOI 10.1090/tran/7005
- Patrick Billingsley, Convergence of probability measures, 2nd ed., Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, Inc., New York, 1999. A Wiley-Interscience Publication. MR 1700749, DOI 10.1002/9780470316962
- V. I. Bogachev, Measure theory. Vol. I, II, Springer-Verlag, Berlin, 2007. MR 2267655, DOI 10.1007/978-3-540-34514-5
- Francis Bonahon, Bouts des variétés hyperboliques de dimension $3$, Ann. of Math. (2) 124 (1986), no. 1, 71–158 (French). MR 847953, DOI 10.2307/1971388
- Francis Bonahon, The geometry of Teichmüller space via geodesic currents, Invent. Math. 92 (1988), no. 1, 139–162. MR 931208, DOI 10.1007/BF01393996
- Francis 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. MR 1105332, DOI 10.1007/978-1-4612-3142-4_{5}
- Martin Bridgeman and Richard D. Canary, Simple length rigidity for Kleinian surface groups and applications, Comment. Math. Helv. 92 (2017), no. 4, 715–750. MR 3718485, DOI 10.4171/CMH/422
- Martin Bridgeman, Richard Canary, and François Labourie, Simple length rigidity for Hitchin representations, Adv. Math. 360 (2020), 106901, 61. MR 4035950, DOI 10.1016/j.aim.2019.106901
- Martin Bridgeman, Richard Canary, François Labourie, and Andres Sambarino, Simple root flows for Hitchin representations, Geom. Dedicata 192 (2018), 57–86. MR 3749423, DOI 10.1007/s10711-017-0305-2
- M. Burger, A. Iozzi, A. Parreau, M. B. Pozzetti: A structure theorem for geodesic currents and length spectrum compactifications, arXiv:1710.07060, 2017.
- Marc Burger, Alessandra Iozzi, Anne Parreau, and Maria Beatrice Pozzetti, The real spectrum compactification of character varieties: characterizations and applications, C. R. Math. Acad. Sci. Paris 359 (2021), 439–463 (English, with English and French summaries). MR 4278899, DOI 10.5802/crmath.123
- Moon Duchin, Christopher J. Leininger, and Kasra Rafi, Length spectra and degeneration of flat metrics, Invent. Math. 182 (2010), no. 2, 231–277. MR 2729268, DOI 10.1007/s00222-010-0262-y
- Viveka Erlandsson and Caglar Uyanik, Length functions on currents and applications to dynamics and counting, In the tradition of Thurston—geometry and topology, Springer, Cham, [2020] ©2020, pp. 423–458. MR 4264584, DOI 10.1007/978-3-030-55928-1_{1}1
- Benson Farb and Dan Margalit, A primer on mapping class groups, Princeton Mathematical Series, vol. 49, Princeton University Press, Princeton, NJ, 2012. MR 2850125
- Joel Friedman, Sheaves on graphs, their homological invariants, and a proof of the Hanna Neumann conjecture: with an appendix by Warren Dicks, Mem. Amer. Math. Soc. 233 (2015), no. 1100, xii+106. With an appendix by Warren Dicks. MR 3289057, DOI 10.1090/memo/1100
- Sa’ar Hersonsky and Frédéric Paulin, On the rigidity of discrete isometry groups of negatively curved spaces, Comment. Math. Helv. 72 (1997), no. 3, 349–388. MR 1476054, DOI 10.1007/s000140050022
- Ilya Kapovich, An integral weight realization theorem for subset currents on free groups, Topology Proc. 50 (2017), 213–236. MR 3580039
- Ilya Kapovich and Tatiana Nagnibeda, Subset currents on free groups, Geom. Dedicata 166 (2013), 307–348. MR 3101172, DOI 10.1007/s10711-012-9797-y
- Alexander S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. MR 1321597, DOI 10.1007/978-1-4612-4190-4
- Igor Mineyev, The topology and analysis of the Hanna Neumann conjecture, J. Topol. Anal. 3 (2011), no. 3, 307–376. MR 2831266, DOI 10.1142/S1793525311000611
- Igor Mineyev, Submultiplicativity and the Hanna Neumann conjecture, Ann. of Math. (2) 175 (2012), no. 1, 393–414. MR 2874647, DOI 10.4007/annals.2012.175.1.11
- Jean-Pierre Otal, Le spectre marqué des longueurs des surfaces à courbure négative, Ann. of Math. (2) 131 (1990), no. 1, 151–162 (French). MR 1038361, DOI 10.2307/1971511
- Walter Rudin, Real and complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1987. MR 924157
- Dounnu Sasaki, An intersection functional on the space of subset currents on a free group, Geom. Dedicata 174 (2015), 311–338. MR 3303055, DOI 10.1007/s10711-014-0019-7
- Peter Scott, Subgroups of surface groups are almost geometric, J. London Math. Soc. (2) 17 (1978), no. 3, 555–565. MR 494062, DOI 10.1112/jlms/s2-17.3.555
- Peter Scott, Correction to: “Subgroups of surface groups are almost geometric” [J. London Math. Soc. (2) 17 (1978), no. 3, 555–565; MR0494062 (58 #12996)], J. London Math. Soc. (2) 32 (1985), no. 2, 217–220. MR 811778, DOI 10.1112/jlms/s2-32.2.217
- Eric L. Swenson, Quasi-convex groups of isometries of negatively curved spaces, Topology Appl. 110 (2001), no. 1, 119–129. Geometric topology and geometric group theory (Milwaukee, WI, 1997). MR 1804703, DOI 10.1016/S0166-8641(99)00166-2