Infinite motion and 2-distinguishability of graphs and groups

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 Aut(X) on X is 2-distinguishable in all but finitely many instances.\ud 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

Infinite Graphs with Finite 2-Distinguishing Cost

“…It says that the vertex stabilizers of graphs G ∈ Γ are finite if Aut(G) is countably infinite. This follows, for example, from the slightly more general Corollary 3.10 from [9]. Below we state and prove a related result that invokes neither Theorem 2.1 nor Theorem 2.2, but needs the following definition: The set of vertices u ∈ V (G) for which d(u, v) n is called ball of radius n centered at v, and denoted B v (n).…”

“…The reverse implication is not hard to show. In [7] it is attributed to [9], but only the consequence that G is 2-distinguishable is mentioned there. As that paper contains no proof of it, we include one here for the sake of completeness.…”

“…For finite graphs it was shown that many infinite families of graphs have the property that all but finitely many members are 2-distinguishable; see [9]. Interestingly, in such cases the size of the smaller color class can be extremely small.…”

Improving upper bounds for the distinguishing index

“…Both numbers, D(G) and χ D (G), have been intensively investigated by many authors in recent years [4,5,6,9,16]. Our investigation was motivated by the renowned result of Nordhaus-Gaddum [18] who proved in 1956 the following lower and upper bounds for the sum of the chromatic numbers of a graph and its complement (actually, the upper bound was first proved by Zykov [22] in 1949).…”

Distinguishing graphs with intermediate growth

“…In the case of infinite locally finite graphs the following connection of motion and distinguishing number has been conjectured by Tucker [16]. While this conjecture is still open in its full generality, it is known to be true for many classes of graphs including trees [17], tree-like graphs [9] and graphs with countable automorphism group [10]. It is also known that the countable random graph has infinite motion as well as distinguishing number 2 [9].…”