Journal of Algebraic Combinatorics

, Volume 41, Issue 1, pp 109–122

Infinite motion and 2-distinguishability of graphs and groups

  • Wilfried Imrich
  • Simon M. Smith
  • Thomas W. Tucker
  • Mark E. Watkins
Article

DOI: 10.1007/s10801-014-0529-2

Cite this article as:
Imrich, W., Smith, S.M., Tucker, T.W. et al. J Algebr Comb (2015) 41: 109. doi:10.1007/s10801-014-0529-2

Abstract

A group \(A\) acting faithfully on a set \(X\) is \(2\)-distinguishable if there is a \(2\)-coloring of \(X\) that is not preserved by any nonidentity element of \(A\), equivalently, if there is a proper subset of \(X\) with trivial setwise stabilizer. The motion of an element \(a \in A\) is the number of points of \(X\) that are moved by \(a\), and the motion of the group \(A\) is the minimal motion of its nonidentity elements. When \(A\) is finite, the Motion Lemma says that if the motion of \(A\) is large enough (specifically at least \(2\log _2 |A|\)), then the action is \(2\)-distinguishable. For many situations where \(X\) has a combinatorial or algebraic structure, the Motion Lemma implies that the action of \(\mathrm{Aut }(X)\) on \(X\) is 2-distinguishable in all but finitely many instances. We prove an infinitary version of the Motion Lemma for countably infinite permutation groups, which states that infinite motion is large enough to guarantee \(2\)-distinguishability. From this, we deduce a number of results, including the fact that every locally finite, connected graph whose automorphism group is countably infinite is \(2\)-distinguishable. One cannot extend the Motion Lemma to uncountable permutation groups, but nonetheless we prove that (under the permutation topology) every closed permutation group with infinite motion has a dense subgroup which is \(2\)-distinguishable. We conjecture an extension of the Motion Lemma which we expect holds for a restricted class of uncountable permutation groups, and we conclude with a list of open questions. The consequences of our results are drawn for orbit equivalence of infinite permutation groups.

Keywords

Distinguishing number Distinguishability Automorphism Infinite graph Infinite permutation group  Motion Orbit-equivalence 

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Wilfried Imrich
    • 1
  • Simon M. Smith
    • 2
  • Thomas W. Tucker
    • 3
  • Mark E. Watkins
    • 4
  1. 1.Montanuniversität LeobenLeobenAustria
  2. 2.Department of Mathematics, NYC College of TechnologyCity University of New YorkNew YorkUSA
  3. 3.Department of MathematicsColgate UniversityHamiltonUSA
  4. 4.Department of MathematicsSyracuse UniversitySyracuseUSA

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