Graphs and Combinatorics

, Volume 5, Issue 1, pp 333–338 | Cite as

A note on bounded automorphisms of infinite graphs

  • Chris D. Godsil
  • Wilfried Imrich
  • Norbert Seifter
  • Mark E. Watkins
  • Wolfgang Woess


LetX be a connected locally finite graph with vertex-transitive automorphism group. IfX has polynomial growth then the set of all bounded automorphisms of finite order is a locally finite, periodic normal subgroup ofAUT(X) and the action ofAUT(X) onV(X) is imprimitive ifX is not finite. IfX has infinitely many ends, the group of bounded automorphisms itself is locally finite and periodic.


Normal Subgroup Automorphism Group Finite Order Polynomial Growth Finite Graph 
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  1. 1.
    Freudenthal, H.: Über die Enden diskreter Räume und Gruppen. Comment. Math. Helv.17, 1–38 (1944)Google Scholar
  2. 2.
    Grigorchuk, R.I.: On the growth degrees ofp-groups and torsion free groups. Math. USSR Sb.,54, 185–205 (1986)Google Scholar
  3. 3.
    Gromov, M.: Groups of polynomial growth and expanding maps. Inst. Hautes Études Sci. Publ. Math.53, 53–78 (1981)Google Scholar
  4. 4.
    Gromov, M.: Infinite groups as geometric objects. In: Proc. International Congress of Mathematicians, Warszawa 1983. pp. 385–392. PWN (1984)Google Scholar
  5. 5.
    Halin, R.: Über unendliche Wege in Graphen. Math. Ann.157, 125–137 (1964)Google Scholar
  6. 6.
    Hopf, H.: Enden offener Räume und unendliche diskontinuierliche Gruppen. Comment. Math. Helv.15, 27–32 (1943)Google Scholar
  7. 7.
    Jung, H.A., Watkins, M.E.: Fragments and automorphisms of infinite graphs. Europ. J. Comb.5, 149–162 (1984)Google Scholar
  8. 8.
    Neumann, B.H.: Groups with finite classes of conjugate elements. Proc. Lond. Math. Soc. III. Ser.1, 178–187 (1951)Google Scholar
  9. 9.
    Sabidussi, G.: Vertex transitive graphs. Mh. Math.68, 385–401 (1964)Google Scholar
  10. 10.
    Schwerdtfeger, H.: Introduction to group theory. Leyden: Noordhoff International Publishing 1976Google Scholar
  11. 11.
    Trofimov, V.I.: Automorphisms of graphs and a characterization of lattices. Math. USSR Izv.22, 379–391 (1984)Google Scholar
  12. 12.
    Trofimov, V.I.: Graphs with polynomial growth. Math. USSR Sb.51, 405–417 (1985)Google Scholar
  13. 13.
    Wolf, J.A.: Growth of finitely generated solvable groups and curvature of Riemannian manifolds. J. Differ. Geom.2, 421–446 (1968)Google Scholar

Copyright information

© Springer-Verlag 1989

Authors and Affiliations

  • Chris D. Godsil
    • 1
  • Wilfried Imrich
    • 2
  • Norbert Seifter
    • 2
  • Mark E. Watkins
    • 3
  • Wolfgang Woess
    • 2
  1. 1.Department of Combinatorics and OptimizationUniversity of WaterlooWaterlooCanada
  2. 2.Institut für Mathematik und Angewandte Geometrie, MontanuniversitätLeobenAustria
  3. 3.Department of MathematicsSyracuse UniversitySyracuseUSA

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