MOVEMENT AND SEPARATION OF SUBSETS OF POINTS UNDER GROUP ACTIONS

Abstract

Let G be a permutation group on a set Ω, and let m and k be integers where 0<m<k. For a subset Γ of Ω, if the cardinalities of the sets Γ^g\Γ, for g∈G, are finite and bounded, then Γ is said to have bounded movement, and the movement of Γ is defined as move (Γ) =max_g∈G[mid ]Γ^g\Γ[mid ]. If there is a k-element subset Γ such that move (Γ)[les ]m, it is shown that some G-orbit has length at most (k²−m)\(k−m). When combined with a result of P. M. Neumann, this result has the following consequence: if some infinite subset Γ has bounded movement at most m, then either Γ is a G-invariant subset with at most m points added or removed, or Γ nontrivially meets a G-orbit of length at most m²+m+1. Also, if move (Γ)[les ]m for all k-element subsets Γ and if G has no fixed points in Ω, then either [mid ]Ω[mid ][les ]k+m (and in this case all permutation groups on Ω have this property), or [mid ]Ω[mid ][les ]5m−2. These results generalise earlier results about the separation of finite sets under group actions by B. J. Birch, R. G. Burns, S. O. Macdonald and P. M. Neumann, and groups in which all subsets have bounded movement (by the author).