Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 


Graphs of hyperbolic groups and a limit set intersection theorem

Author: Pranab Sardar
Journal: Proc. Amer. Math. Soc.
MSC (2010): Primary 20F67
Published electronically: December 26, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We define the notion of limit set intersection property for a collection of subgroups of a hyperbolic group; namely, for a hyperbolic group $ G$ and a collection of subgroups $ \mathcal S$ we say that $ \mathcal S$ satisfies the limit set intersection property if for all $ H,K \in \mathcal S$ we have $ \Lambda (H)\cap \Lambda (K)=\Lambda (H\cap K)$. Given a hyperbolic group admitting a decomposition into a finite graph of hyperbolic groups structure with QI embedded condition, we show that the set of conjugates of all the vertex and edge groups satisfies the limit set intersection property.

References [Enhancements On Off] (What's this?)

  • [Anda] James W. Anderson, The limit set intersection theorem for finitely generated Kleinian groups, Math. Res. Lett. 3 (1996), no. 5, 675-692. MR 1418580,
  • [Andb] James W. Anderson, Limit set intersection theorems for Kleinian groups and a conjecture of Susskind, Comput. Methods Funct. Theory 14 (2014), no. 2-3, 453-464. MR 3265372,
  • [BF92] M. Bestvina and M. Feighn, A combination theorem for negatively curved groups, J. Differential Geom. 35 (1992), no. 1, 85-101. MR 1152226
  • [BF96] Mladen Bestvina and Mark Feighn, Addendum and correction to: ``A combination theorem for negatively curved groups'' [J. Differential Geom. 35 (1992), no. 1, 85-101; MR1152226 (93d:53053)], J. Differential Geom. 43 (1996), no. 4, 783-788. MR 1412684
  • [BH99] Martin R. Bridson and André Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. MR 1744486
  • [Ger98] S. M. Gersten, Cohomological lower bounds for isoperimetric functions on groups, Topology 37 (1998), no. 5, 1031-1072. MR 1650363,
  • [GMRS97] Rita Gitik, Mahan Mitra, Eliyahu Rips, and Michah Sageev, Widths of subgroups, Trans. Amer. Math. Soc. 350 (1998), no. 1, 321-329. MR 1389776,
  • [Gro85] M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75-263. MR 919829,
  • [Hat01] Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. MR 1867354
  • [Kap00a] Ilya Kapovich, Mapping tori of endomorphisms of free groups, Comm. Algebra 28 (2000), no. 6, 2895-2917. MR 1757436,
  • [Kap00b] Ilya Kapovich, A non-quasiconvex subgroup of a hyperbolic group with an exotic limit set, New York J. Math. 1 (1994/95), 184-195, electronic. MR 1362975
  • [Kap01] Ilya Kapovich, The combination theorem and quasiconvexity, Internat. J. Algebra Comput. 11 (2001), no. 2, 185-216. MR 1829050,
  • [Mit98] Mahan Mitra, Cannon-Thurston maps for trees of hyperbolic metric spaces, J. Differential Geom. 48 (1998), no. 1, 135-164. MR 1622603
  • [Ser00] Jean-Pierre Serre, Trees, translated by corrected 2nd printing of the 1980 English translation translated from the French original by John Stillwell, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. MR 1954121
  • [SS92] Perry Susskind and Gadde A. Swarup, Limit sets of geometrically finite hyperbolic groups, Amer. J. Math. 114 (1992), no. 2, 233-250. MR 1156565,
  • [SW79] Peter Scott and Terry Wall, Topological methods in group theory, Homological group theory (Proc. Sympos., Durham, 1977) London Math. Soc. Lecture Note Ser., vol. 36, Cambridge Univ. Press, Cambridge-New York, 1979, pp. 137-203. MR 564422
  • [Yan12] Wen-yuan Yang, Limit sets of relatively hyperbolic groups, Geom. Dedicata 156 (2012), 1-12. MR 2863542,

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 20F67

Retrieve articles in all journals with MSC (2010): 20F67

Additional Information

Pranab Sardar
Affiliation: Indian Institute of Science Education and Research Mohali, Knowledge City, Sector 81, SAS Nagar, Manauli P.O. 140306, India

Keywords: Hyperbolic groups, limit sets, Bass-Serre theory
Received by editor(s): September 13, 2016
Received by editor(s) in revised form: June 27, 2017
Published electronically: December 26, 2017
Communicated by: Ken Bromberg
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society