English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

A group A acting faithfully on a set X is 2-distinguishable if there is a2-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 bya, and the motion of the group A is the minimal motion of its nonidentityelements. For finite A, the Motion Lemma says that if the motion of A is largeenough (specifically at least 2 log_2 |A|), then the action is2-distinguishable. For many situations where X has a combinatorial or algebraicstructure, the Motion Lemma implies the action of Aut(X) on X is2-distinguishable in all but finitely many instances.  We prove an infinitary version of the Motion Lemma for countably infinitepermutation groups, which states that infinite motion is large enough toguarantee 2-distinguishability. From this we deduce a number of results,including the fact that every locally finite, connected graph whoseautomorphism group is countably infinite is 2-distinguishable. One cannotextend the Motion Lemma to uncountable permutation groups, but nonetheless weprove that 2-distinguishable permutation groups with infinite motion are densein the class of groups with infinite motion. We conjecture an extension of theMotion Lemma which we expect holds for a restricted class of uncountablepermutation groups, and we conclude with a list of open questions. Theconsequences of our results are drawn for orbit equivalence of infinitepermutation groups.

Infinite Motion and 2-Distinguishability of Graphs and Groups