Imrich, Wilfried, Smith, Simon M., Tucker, Thomas W. et al and Watkins, Mark E.
(2015)
Infinite motion and 2distinguishability of graphs and groups.
Journal of Algebraic Combinatorics, 41
(1).
pp. 109122.
ISSN 09259899
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Abstract
A group A acting faithfully on a set X is 2distinguishable if there is a 2coloring 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 2distinguishable. For many situations where X has a combinatorial or algebraic structure, the Motion Lemma implies that the action of Aut(X) on X is 2distinguishable 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 2distinguishability. From this we deduce a number of results, including the fact that every locally finite, connected graph whose automorphism group is countably infinite is 2distinguishable. 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 2distinguishable. 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.
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