Skip to main content
SpringerLink
Log in

Abstract homomorphisms from some topological groups to acylindrically hyperbolic groups

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

We describe homomorphisms \(\varphi :H\rightarrow G\) for which G is acylindrically hyperbolic and H is a topological group which is either completely metrizable or locally countably compact Hausdorff. It is shown that, in a certain sense, either the image of \(\varphi \) is small or \(\varphi \) is almost continuous. We also describe homomorphisms from the Hawaiian earring group to G as above. We prove a more precise result for homomorphisms \(\varphi :H\rightarrow {\text {Mod}}(\Sigma )\), where H is as above and \({\text {Mod}}(\Sigma )\) is the mapping class group of a connected compact surface \(\Sigma \). In this case there exists an open normal subgroup \(V\leqslant H\) such that \(\varphi (V)\) is finite. We also prove the analogous statement for homomorphisms \(\varphi :H\rightarrow {\text {Out}}(G)\), where G is a one-ended hyperbolic group. Some automatic continuity results for relatively hyperbolic groups and fundamental groups of graphs of groups are also deduced. As a by-product, we prove that the Hawaiian earring group is acylindrically hyperbolic, but does not admit any universal acylindrical action on a hyperbolic space.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Abbott, C., Balasubramanya, S., Osin, D.: Hyperbolic structures on groups. Algebr. Geom. Topol. 19, 1747–1835 (2019)

    Article  MathSciNet  Google Scholar 

  2. Abbot, C.R.: Not all finitely generated groups have universal acylindrical actions. Proc. Amer. Math. Soc. 144(10), 4151–4155 (2016)

  3. Abbott, C.R., Behrstock J., Durham, M.C.: Largest acylindrical actions and stability in hierarchically hyperbolic groups. Trans. Am. Math. Soc. Ser. B 8, 66–104 (2021)

  4. Bader, U., Furman, A.: Margulis’ super-rigidity theorem for non-lattices (2018). arXiv:1810.01608v1

  5. Behrstock, J., Hagen, M.F., Sisto, A.: Hierarchically hyperbolic spaces II: combination theorems and the distance formula (2015). arXiv: 1509.0063v2

  6. Behrstock, J., Hagen, M.F., Sisto, A.: Hierarchically hyperbolic spaces I: curve complexes for cubical groups. Geom. Topol. 21, 1731–1804 (2017)

  7. Birman, J.S., Chillingworth, D.R.J.: On the homeotopy group of a non-orientable surface. Proc. Camb. Philos. Soc. 71, 437–448 (1972)

    Article  MathSciNet  Google Scholar 

  8. Bogopolski, O.: Equations in acylindrically hyperbolic groups and verbal closedness (2019). Accepted to Groups, Geometry and Dynamics. Available at https://arxiv.org/pdf/1805.08071.pdf

  9. Bogopolski, O., Bux, K.-U.: From local to global conjugacy of subgroups of relatively hyperbolic groups. Intern. J. Algebra Comput. 27(3), 299–314 (2017)

  10. Bogopolskii, O.V., Gerasimov, V.N.: Finite subgroups of hyperbolic groups, Algebra Logic 34, 343–345 (1996)

  11. Bowditch, B.: Cut points and canonical splittings of hyperbolic groups. Acta. Math. 180, 145–186 (1998)

    Article  MathSciNet  Google Scholar 

  12. Bowditch, B.: Intersection numbers and the hyperbolicity of the curve complex. J. Reine Angew. Math. 598, 105–129 (2006)

  13. Bowditch, B.: Tight geodesics in the curve complex. Invent. Math. 171, 281–300 (2008)

    Article  MathSciNet  Google Scholar 

  14. Braun, O., Hofmann, K.H., Kramer, L.: Automatic continuity of abstract homomorphisms between locally compact and polish groups. Transform. Groups 25, 1–32 (2020)

    Article  MathSciNet  Google Scholar 

  15. Bridson, M., Haeffliger, A.: Metric Spaces of Non-positive Curvature. Springer, Berlin (1999)

    Book  Google Scholar 

  16. Cannon J., Conner, G.: The combinatorial structure of the Hawaiian earring group. Topol. Appl. 106, 225–271 (2000)

  17. Conner, G., Corson, S.: A note on automatic continuity. Proc. Am. Math. Soc. 147, 1255–1268 (2019)

    Article  MathSciNet  Google Scholar 

  18. Conner G., Spencer K.: Anomalous behavior of the Hawaiian Earring group. J. Group Theory 8, 225–227 (2005)

  19. Coornaert, M., Delzant, T., Papadopoulos, A.: Geometrie et theorie des groupes. Les groupes hyperboliques de Gromov. Lecture Notes in Mathematics, vol 1441, x+165 pp. Springer, Berlin (1990)

  20. Corson S., Kazachkov I.: On preservation of automatic continuity. Monatsh. Math. 191, 37–52 (2020)

  21. Coulon, R.: On the geometry of Burnside quotients of torsion free hyperbolic groups. Int. J. Algebra Comput. 24(3), 251–345 (2014)

  22. Dahmani, F., Guirardel, V., Osin, D.: Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces. Mem. Am. Math. Soc. 245, 1156 (2016)

    MathSciNet  MATH  Google Scholar 

  23. Dudley, R.: Continuity of homomorphisms. Duke Math. J. 28, 587–594 (1961)

  24. Eda, K.: Free \(\alpha \)-products and fundamental groups of subspaces of the plane. Topol. Appl. 84, 283–306 (1998)

  25. Eda, K.: Atomic property of the fundamental groups of the Hawaiian earring and wild locally path-connected spaces. J. Math. Soc. Jpn. 63, 769–787 (2011)

    Article  MathSciNet  Google Scholar 

  26. Eda, K., Kawamura, K.: The singular homology of the Hawaiian earring. J. Lond. Math. Soc. 62, 305–310 (2000)

    Article  MathSciNet  Google Scholar 

  27. Farb, B., Margalit, D.: A Primer on Mapping Class Groups. Princeton Mathematical Series, vol. 49. Princeton University Press, Princeton (2011)

  28. Farb, B., Masur, H.: Superrigidity and mapping class groups. Topology 37, 1169–1176 (1998)

  29. Fujiwara, K.: On the outer automorphism group of a hyperbolic group. Israel J. Math. 131, 277–284 (2002)

    Article  MathSciNet  Google Scholar 

  30. Griffiths, H.: Infinite products of semi-groups and local connectivity. Proc. Lond. Math. Soc. 3, 455–456 (1956)

    Article  MathSciNet  Google Scholar 

  31. Gromov, M.: Hyperbolic groups. In: Essays in Group Theory. Springer, New York, pp.75–263 (1987)

  32. Harvey, W.J.: Boundary structure of the modular group. In: Riemann Surfaces and Related Topics, Proceedings of the 1979 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978), v. 97 of Ann. of Math. Stud., pages 245-251, Princeton, N.J., (1981). Princeton Univ. Press

  33. Harvey, W.J., Maclachlan, C.: On mapping class groups and Teichmüller spaces. Proc. Lond. Math. Soc. 30, 496–512 (1975)

    MATH  Google Scholar 

  34. Higman, G.: Unrestricted free products and varieties of topological groups. J. Lond. Math. Soc. 27, 73–81 (1952)

    Article  MathSciNet  Google Scholar 

  35. Hrušák, M., van Mill, J., Ramos-García, U., Shelah, S.: Countably compact groups without non-trivial convergent sequences. Trans. Am. Math. Soc. V 374, 1277–1296 (2021)

    Article  MathSciNet  Google Scholar 

  36. Ivanov, N.V.: Subgroups of Teichmüller modular groups. Vol. 115 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, Translated from the Russian by E.J.F. Primrose and revised by the author. (1992)

  37. Klyachko, A.A., Ol’shanskii, A.Yu., Osin, D.V.: On topologizable and non-topologizable groups. Topology Appl. 160, 2104–2120 (2013)

  38. Koubi, M.: Croissance uniforme dans les groupes hyperboliques. Ann. Inst. Fourier (Grenoble) 48(5), 1441–1453 (1998)

  39. Kramer, L., Varghese, O.: Abstract homomorphisms from locally compact groups to discrete groups. J. Algebra 538, 127–139 (2019)

    Article  MathSciNet  Google Scholar 

  40. Levitt, G.: Automorphisms of hyperbolic groups and graph of groups. Geom. Dedicata 114, 49–70 (2005)

    Article  MathSciNet  Google Scholar 

  41. Margulis, G.A.: Discrete Subgroups of Semisimple Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 17. Springer, Berlin (1991)

  42. Masur, H.A., Minsky, Y.N.: Geometry of the complex of curves I. Hyperbolicity. Invent. Math. 138(1), 103–149 (1999)

  43. Morris, S.A., Nickolas, P.: Locally compact group topologies on an algebraic free product of groups. J. Algebra 38(2), 393–397 (1976)

  44. Nakamura, J.: Atomic properties of the Hawaiian earring group for HNN extensions. Commun. Algebra 43, 4138–4147 (2015)

  45. Nikolov, N., Segal, D.: On finitely generated profinite groups, I: strong completeness and uniform bounds. Ann. Math. 165, 171–238 (2007)

    Article  MathSciNet  Google Scholar 

  46. Osin, D.: Groups acting acylindrically on hyperbolic spaces. In: Proc. ICM-2018, Rio de Janeiro. Vol. 1, pp. 915–936, World Scientic (2018); Also available at arXiv:1712.00814

  47. Osin, D.: Relatively hyperbolic groups: Intrinsic geometry, algebraic properties, and algorithmic problems. Mem. Am. Math. Soc. 179(843) (2006)

  48. Osin, D.: Acylindrically hyperbolic groups. Trans. Am. Math. Soc. 368, 851–888 (2016)

    Article  MathSciNet  Google Scholar 

  49. Paolini, G., Shelah, S.: Polish topologies for graph products of groups. J. Lond. Math. Soc. 100, 383–403 (2019)

    Article  MathSciNet  Google Scholar 

  50. Rosendal, C.: Completely metrisable groups acting on trees. J. Symb. Log. 76, 1005–1022 (2011)

  51. Serre, J.P.: Trees. Springer, Berlin (1980)

    Book  Google Scholar 

  52. Sisto, A.: Contracting elements and random walks. Journal für die reine und angewandte Mathematik (Crelles Journal) 742, 79–114 (2018)

  53. Stroppel, M.: Locally compact groups. EMS textbooks in Math, EMS (2006)

  54. Szepietowski, B.: Embedding the braid group in mapping class groups. Publ. Mat. 54, 359–368 (2010)

  55. Thurston, W.P.: On the geometry and dynamics of diffeomorphisms of surfaces. Bull Am. Math. Soc. (NS) 19(2), 417–431 (1988)

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, University of Szczecin, ul. Wielkopolska 15, 70-451, Szczecin, Poland

    Oleg Bogopolski

  2. Düsseldorf University, Düsseldorf, Germany

    Oleg Bogopolski

  3. School of Mathematics, University of Bristol, Fry Building, Woodland Road, Bristol, BS8 1UG, UK

    Samuel M. Corson

Authors
  1. Oleg Bogopolski
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Samuel M. Corson
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by Christian Rosendal.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The work of the second author is supported by European Research Council grant PCG-336983 and by the Severo Ochoa Programme for Centres of Excellence in R&D SEV-20150554.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bogopolski, O., Corson, S.M. Abstract homomorphisms from some topological groups to acylindrically hyperbolic groups. Math. Ann. 384, 1017–1055 (2022). https://doi.org/10.1007/s00208-021-02278-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-021-02278-4

Mathematics Subject Classification

Search

Navigation