LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-22

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Finite stabilizers in Aut(Q,<) are extremely amenable

Statement

$Aut (Q, <)$ , with the topology of pointwise convergence, is extremely amenable, and so is the pointwise stabiliser of every finite subset of $Q$ : every continuous action of such a group on a nonempty compact Hausdorff space has a fixed point.

Facts & Assumptions

Given: A finite subset $E \subseteq Q$ and a continuous action of $fix (E)$ on a nonempty compact Hausdorff space.

[F1]

Finite Ramsey for colourings of $k$ -element subsets: for all positive $k, c, r$ there is $N$ with $N \to (r)_{c}^{k}$ (For positive $k, c, r$ there is an $N$ such that every $c$ -colouring of $[N]^{k}$ has a monochromatic $r$ -element set, Finite colourings of $k$ -element subsets, monochromatic sets, and the arrow notations $N \to (s, t)^{2}$ and $N \to (r)_{c}^{k}$ , The natural numbers $N$ (von Neumann)).

[F2]

The KPT correspondence: for a Fraïssé structure with rigid finite substructures and the Ramsey property, the automorphism group is extremely amenable. This is Kechris--Pestov--Todorcevic, Theorem 4.7; its finite-linear- order instance is the one used here. The Ramsey hypothesis for that instance, but not the KPT fixed-point conclusion itself, is supplied by For positive $k, c, r$ there is an $N$ such that every $c$ -colouring of $[N]^{k}$ has a monochromatic $r$ -element set.

[L1]

A finite point stabiliser of $Aut (Q, <)$ is the direct product of the automorphism groups of the finitely many open intervals cut out by the support, each of which is order-isomorphic to $Q$ ; a finite product of extremely amenable groups is extremely amenable, because fixed points can be taken one factor at a time: an action of $G_{1} \times G_{2}$ on a compact space has a fixed point for $G_{1}$ by extreme amenability of $G_{1}$ , the fixed-point set is compact and invariant under $G_{2}$ , and extreme amenability of $G_{2}$ supplies a point fixed by both. [given]

Proof

technique · direct

1.1

The age of $(Q, <)$ consists of the finite linear orders, each of which is rigid, and the Ramsey property required by the KPT correspondence is precisely finite Ramsey for colourings of $k$ -element subsets, since a colouring of embeddings of the $a$ -element order into an $N$ -element order is a colouring of $a$ -element subsets of $N$ and a homogeneous $b$ -element subset is a monochromatic copy.

1.2

For the stabiliser of a finite $E$ : the points of $E$ cut $Q$ into finitely many open intervals, each order-isomorphic to $Q$ , and $fix (E)$ is the direct product of the automorphism groups of those intervals.

givenL1

2.1

By [F2] applied to the age of $(Q, <)$ described in step 1.1, $Aut (Q, <)$ is extremely amenable.

step 1.1F2

3.1

By step 2.1 and [L1], applied factor by factor to the finitely many interval automorphism groups of step 1.2, $fix (E)$ is extremely amenable: the fixed-point set of an action is computed one factor at a time, and each factor contributes a fixed point because it is an automorphism group of a copy of $Q$ and hence extremely amenable by step 2.1.

step 2.1step 1.2L1

4.1

Thus $Aut (Q, <)$ and each of its finite point stabilisers is extremely amenable, which is the assertion of the statement; the argument used the ZF theorem of [F1] and the fixed-point criterion of [F2] only.

step 2.1step 3.1F1F2∎

Remarks

Why the finite-linear-order instance suffices here. The permutation model of this pair uses only the rational-ordered atom set of Brunner's ordered Läuchli permutation models, whose automorphism group is $Aut (Q, <)$ ; the finite stabiliser form of the statement is what the BPI theorem consumes.
The product argument is where "finite" is used. A finite product of extremely amenable groups is extremely amenable by the one-factor-at-a-time argument; an infinite product need not be, and no such claim is made.

Depends on

Used by

Relative consistency of BPI without Urysohn's lemma Theorem

Dependency tree · two levels

32 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Kechris, Pestov, and Todorcevic, Fraïssé limits, Ramsey theory, and topological dynamics of automorphism groups (standard reference, not scraped)
Andreas Blass, Partitions and Permutation Groups (standard reference, not scraped)