Locally Moving Groups and the Reconstruction Problem for Chains and Circles

For groups of automorphisms of chains (linear orders) and circles which satisfy certain weak transitivity hypotheses, the Dedekind completion of the chain or circle can be reconstructed up to reversal of order or orientation from the group alone. Let 〈L,<〉 be a dense chain. Throughout, an interval means a nonempty open interval. A subgroup G ≤ Aut(〈L,<〉) is locally moving if for every interval I there exists g ∈ G − {Id} such that g↾(L− I) = Id . G is n-interval-transitive if for every pair of sequences of intervals I1 < . . . < In and J1 < . . . < Jn there exists g ∈ G such that g(I1) ∩ J1 6= ∅, . . . , g(In) ∩ Jn 6= ∅. The Dedekind completion of L is denoted by L̄. It is assumed that L̄ does not add endpoints to L. Theorem 1 Assume Li has no endpoints and let Gi ≤ Aut(Li) be locally moving and 2-interval-transitive, i = 1, 2. Suppose that α : G1 ∼= G2. Then there is a monotonic bijection τ : L̄1 → L̄2 which induces α, that is, α(g) = τ ◦ g ◦ τ−1 for every g ∈ G; and τ is unique. (In particular, if G ≤ Aut(L) is locally moving and 2-interval-transitive, then every α ∈ Aut(G) is conjugation by a unique monotonic bijection τ of L̄.) The next theorem shows that the hypothesis of 2-interval-transitivity in Theorem 1 is rather weak. Theorem 2 There exist L without endpoints and G ≤ Aut(L) such that G is locally moving and 2-interval-transitive, but (i) G is not 3-interval-transitive, and (ii) G does not act 2-order-transitively on any of its orbits in L̄ (that is, there is no orbit L0 ⊆ L̄ such that for every x < y and u < v in L0 there exists g ∈ G such that g(x) = u and g(y) = v). The “circularization” of L is the structure CR (L) = 〈L,Cr〉, where Cr(x, y, z) ≡ (x < y < z) ∨ (y < z < x) ∨ (z < x < y). For pairwise disjoint circular intervals I1, . . . , In, Cr(I1, . . . , In) means that for every 1 ≤ i < j < k ≤ n and x, y, z in Ii, Ij, Ik respectively, Cr(x, y, z) holds. n-interval-transitivity is defined here as for chains, but with “I1 < . . . < In” replaced by “Cr(I1, . . . , In)”. Theorem 3 Let Gi ≤ Aut(CR (Li)) be locally moving and 3-interval-transitive, i = 1, 2. Suppose that α : G1 ∼= G2. Then there is a unique orientation-preserving or -reversing bijection τ : CR (L̄1) → CR (L̄2) which induces α. Theorem 4 In Theorems 1 and 3 there is a first order interpretation of the action of G on a dense subset of L̄ (respectively, CR (L̄)).