Relative second bounded cohomology of free groups

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• 作者：Cristina Pagliantini (1)
  Pascal Rolli (1)
  
  1. Department Mathematik ; ETH Z眉rich ; R盲mistrasse 101 ; 8092聽 ; Zurich ; Switzerland
  
• 关键词：Relative bounded cohomology ; Quasimorphisms ; Free groups ; Split quasimorphisms ; Schreier graphs ; 20J06 ; 20E05 ; 55N10 ; 57M15
• 刊名：Geometriae Dedicata
• 出版年：2015
• 出版时间：April 2015
• 年：2015
• 卷：175
• 期：1
• 页码：267-280
• 全文大小：225 KB
• 参考文献：1. Bowditch, B.H.: Relatively hyperbolic groups. Int. J. Algebra Comput. 22(3), 1250016, 66 (2012)
  2. Brooks, R.: Some remarks on bounded cohomology. In: Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference. State Univ. New York, Stony Brook, NY (1978); Ann. Math. Stud. vol. 97, pp. 53鈥?3. Princeton Univ. Press, Princeton, NJ (1981)
  3. Bestvina, M., Bromberg, K., Fujiwara, K.: Bounded Cohomology Via Quasi-trees. arXiv:1306.1542 (2013)
  4. Burger, M, Monod, N (1999) Bounded cohomology of lattices in higher rank Lie groups. J. Eur. Math. Soc. 1: pp. 199-235 CrossRef
  5. Burger, M, Monod, N (2002) Continuous bounded cohomology and applications to rigidity theory. Geom. Funct. Anal. 12: pp. 219-280 CrossRef
  6. Calegari, D.: scl, Mathematical Society of Japan, (2009)
  7. Dahmani, F., Guirardel, V., Osin, D.: Hyperbolically Embedded Subgroups and Rotating Families in Groups Acting on Hyperbolic Spaces. arXiv:1111.7048 (2011)
  8. Frigerio, R, Pagliantini, C (2012) Relative measure homology and continuous bounded cohomology of topological pairs. Pac. J. Math. 257: pp. 91-130 CrossRef
  9. Frigerio, R., Pozzetti, B., Sisto, A.: Extending Higher Dimensional Quasi-Cocycles. arXiv:1311.7633 (2013)
  10. Gromov, M.: Volume and bounded cohomology. Inst. Hautes 脡tudes Sci. Publ. Math. (1982), (56), 5鈥?9 (1983)
  11. Gromov, M.: Hyperbolic groups. In: Gersten, S.M. (ed.) 鈥淓ssay in Group Theory鈥?Math. Sci. Res. Inst. Publ., Springer, New York, no. 8, pp. 75鈥?63 (1987)
  12. Hull, M., Osin, D.: Induced quasicocycles on groups with hyperbolically embedded subgroups. Algebr. Geom. Topol. (13), 2635鈥?665 (2013)
  13. Ivanov, N.V.: Foundations of the theory of bounded cohomology, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), vol. 143, pp. 69鈥?09, pp. 177鈥?78, Studies in Topology, V (1985)
  14. Ivanov, NV (1988) Second bounded cohomology group. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 167: pp. 117-120
  15. L枚h, C.: Simplicial volume. Bull. Man. Atl., pp. 7鈥?8, (2011). http://www.map.mpim-bonn.mpg.de/Simplicial_volume
  16. Matsumoto, S, Morita, S (1985) Bounded cohomology of certain groups of homeomorphisms. Proc. Am. Math. Soc. 94: pp. 539-544 CrossRef
  17. Mineyev, I (2001) Straightening and bounded cohomology of hyperbolic groups. GAFA Geom. Funct. Anal. 11: pp. 807-839 CrossRef
  18. Mineyev, I (2002) Bounded cohomology characterizes hyperbolic groups. Q. J. Math. Oxf. Ser. 53: pp. 59-73 CrossRef
  19. Mitsumatsu, Y (1984) Bounded cohomology and $$l^1$$ l 1 -homology of surfaces. Topology 23: pp. 465-471 CrossRef
  20. Monod, N.: Continuous Bounded Cohomology of Locally Compact Groups, Lecture Notes in Mathematics, vol. 1758. Springer, Berlin (2001)
  21. Pagliantini, C.: Relative (Continuous) Bounded Cohomology and Simplicial Volume of Hyperbolic Manifolds with Geodesic Boundary. PhD thesis, Universit脿 di Pisa (2012). http://etd.adm.unipi.it/theses/available/etd-07112012-101103/
  22. Park, HS (2003) Relative bounded cohomology. Topol. Appl. 131: pp. 203-234 CrossRef
  23. Rolli, P.: Quasi-Morphisms on Free Groups. arXiv:0911.4234 (2009)
  24. Rolli, P.: Split Quasicocycles. arXiv:1305.0095 (2013)
  25. Rolli, P.: Split Quasicocycles and Defect Spaces. PhD thesis, ETH Zurich (2014). doi:10.3929/ethz-a-010168196
  26. Stallings, JR (1983) Topology of finite graphs. Invent. Math. 71: pp. 551-565 CrossRef
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Geometry
  
• 出版者：Springer Netherlands
• ISSN：1572-9168

文摘

This paper is devoted to the computation of the space \(\mathrm{H}_\mathrm{b}^2(\Gamma ,H;{\mathbb {R}})\) , where \(\Gamma \) is a free group of finite rank \(n\ge 2\) and \(H\) is a subgroup of finite rank. More precisely we prove that \(H\) has infinite index in \(\Gamma \) if and only if \(\mathrm{H}_\mathrm{b}^2(\Gamma ,H;{\mathbb {R}})\) is not trivial, and furthermore, if and only if there is an isometric embedding \(\oplus _\infty ^n\mathcal {D}({\mathbb {Z}})\hookrightarrow \mathrm{H}_\mathrm{b}^2(\Gamma ,H;{\mathbb {R}})\) , where \(\mathcal {D}({\mathbb {Z}})\) is the space of bounded alternating functions on \({\mathbb {Z}}\) equipped with the defect norm.