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ON THE ACTIONS OF HIGMAN-THOMPSON GROUPS BY HOMEOMORPHISMS

  • Kim, Jin Hong (Department of Mathematics Education Chosun University)
  • Received : 2019.03.21
  • Accepted : 2019.07.26
  • Published : 2020.03.31

Abstract

The aim of this short paper is to show some rigidity results for the actions of certain finitely presented groups by homeomorphisms. As an interesting and special case, we show that the actions of Higman-Thompson groups by homeomorphisms on a cohomology manifold with a non-zero Euler characteristic should be trivial. This is related to the wellknown Zimmer program and shows that the actions by homeomorphism could be very much different from those by diffeomorphisms.

Acknowledgement

Supported by : National Research Foundation of Korea (NRF)

References

  1. G. Arzhantseva, M. Bridson, T. Januszkiewicz, I. Leary, A. Minasyan, and J. Swiatkowski, Infinite groups with fixed point properties, Geom. Topol. 13 (2009), no. 3, 1229-1263. https://doi.org/10.2140/gt.2009.13.1229
  2. A. Borel, Seminar on Transformation Groups, With contributions by G. Bredon, E. E. Floyd, D. Montgomery, R. Palais. Annals of Mathematics Studies, No. 46, Princeton University Press, Princeton, NJ, 1960.
  3. G. E. Bredon, Sheaf Theory, second edition, Graduate Texts in Mathematics, 170, Springer-Verlag, New York, 1997. https://doi.org/10.1007/978-1-4612-0647-7
  4. M. R. Bridson and K. Vogtmann, Actions of automorphism groups of free groups on homology spheres and acyclic manifolds, Comment. Math. Helv. 86 (2011), no. 1, 73-90. https://doi.org/10.4171/CMH/218
  5. D. Fisher, Groups acting on manifolds: around the Zimmer program, in Geometry, rigidity, and group actions, 72-157, Chicago Lectures in Math, Univ. Chicago Press, Chicago, IL, 2011. https://doi.org/10.7208/chicago/9780226237909.001.0001
  6. G. Higman, Finitely presented infinite simple groups, Department of Pure Mathematics, Department of Mathematics, I.A.S. Australian National University, Canberra, 1974.
  7. K. Kawakubo, The Theory of Transformation Groups, translated from the 1987 Japanese edition, The Clarendon Press, Oxford University Press, New York, 1991.
  8. L. N. Mann and J. C. Su, Actions of elementary p-groups on manifolds, Trans. Amer. Math. Soc. 106 (1963), 115-126. https://doi.org/10.2307/1993717
  9. C. E. Rover, Constructing finitely presented simple groups that contain Grigorchuk groups, J. Algebra 220 (1999), no. 1, 284-313. https://doi.org/10.1006/jabr.1999.7898
  10. D. Witte, Arithmetic groups of higher Q-rank cannot act on 1-manifolds, Proc. Amer. Math. Soc. 122 (1994), no. 2, 333-340. https://doi.org/10.2307/2161021
  11. S. Ye, Euler characteristics and actions of automorphism groups of free groups, Algebr. Geom. Topol. 18 (2018), no. 2, 1195-1204. https://doi.org/10.2140/agt.2018.18.1195