SAMUEL COMPACTIFICATIONS OF AUTOMORPHISM GROUPS
DANA BARTOŠOVÁ AND ANDY ZUCKER

arXiv:1802.02513v1 [math.DS] 7 Feb 2018

1. Introduction
In this paper, we are interested in the automorphism groups of countable first-order structures and the Samuel compactifications of these groups. We will address a variety of questions about the algebraic structure of the Samuel compactification and exhibit connections
between this algebraic structure and the combinatorics of the first-order structures at hand.
Let G be a topological group; all topological groups and spaces will be assumed Hausdorff.
The group G comes with a natural uniform structure, the left uniformity, whose entourages
are of the form {(g, h) ∈ G × G : g −1 h ∈ V } where V ranges over open symmetric neighborhoods of the identity. Every uniform space U admits a Samuel comactifiation, the Gelfand
space of the algebra of bounded uniformly continuous functions on U (see [Sa] or [U]). We
denote by S(G) the Samuel compactification of the group G with its left uniform structure.
In addition to being a compact Hausdorff space, the space S(G) can also be endowed with
algebraic structure. A G-flow is a compact Hausdorff space X equipped with a continuous
right G-action a : X × G → X. Typically the action a is understood, and we write x · g or
xg for a(x, g). We can give S(G) the structure of a G-flow; indeed, for each g ∈ G, the rightmultiplication map h → hg is left-uniformly continuous, so can be continuously extended
to S(G). With some extra work, it can be shown that the evaluation S(G) × G → S(G) is
continuous.
If X and Y are G-flows, a G-map is a continuous map ϕ : X → Y which respects the
G-action. A G-ambit is a pair (X, x0 ), where X is a G-flow and x0 ∈ X has a dense orbit. If
(X, x0 ) and (Y, y0 ) are ambits, then a map of ambits is a G-map ϕ : X → Y with ϕ(x0 ) = y0 .
Notice that there is at most one map of ambits from (X, x0 ) to (Y, y0 ). By identifying G as
embedded into S(G) and by considering the orbit of 1G , we turn (S(G), 1G ) into an ambit.
It turns out that this is the greatest ambit; for any G-ambit (X, x0 ), there is a map of ambits
ϕ : (S(G), 1G ) → (X, x0 ).
We can use this universal property to endow S(G) with yet more structure. A compact
left-topological semigroup is a semigroup S with a compact Hausdorff topology in which the
left multiplication maps t → st are continuous for each s ∈ S. Now let x ∈ S(G); then the
pair (x · G, x) is a G-ambit, so there is a unique G-map λx : S(G) → x · G with λx (1G ) = x.
We can endow S(G) with the structure of a compact left-topological semigroup by setting
xy := λx (y). It is not hard to show that this operation is associative.
Another consequence of the universal property is the existence of universal minimal flows.
Let X be a G-flow. A subflow is any closed Y ⊆ X which is invariant under the G-action.
The G-flow X is minimal if X 6= ∅ and every orbit is dense; equivalently, X is minimal if X
contains no proper subflows. An easy Zorn’s Lemma argument shows that every flow contains
a minimal subflow. The G-flow X is universal if there is a G-map from X onto any minimal
flow. Let M ⊆ S(G) be any minimal subflow, and let Y be a minimal flow. Pick y0 ∈ Y
arbitrarily, making (Y, y0 ) an ambit. Then there is a map of ambits ϕ : (S(G), 1G ) → (Y, y0 ),
1

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D. BARTOŠOVÁ AND A. ZUCKER

and ϕ|M : M → Y is a G-map. We have just shown the existence of a universal minimal
flow. By using some techniques from the theory of compact left-topological semigroups,
it can be shown that there is a unique universal minimal flow up to G-flow isomorphism,
denoted M (G).
The existence and uniqueness of the universal minimal flow suggests another “canonical”
G-ambit we can construct. If X is a G-flow, we can view each g ∈ G as the function
ρg : X → X. Form the product space X X , and set E(X) = {ρg : g ∈ G}. It will be useful in
this instance to write functions on the right, so if f ∈ X X , we write x·f or xf instead of f (x).
The group G acts on E(X) via x · (f · g) = (x · f ) · g. Notice that ρg · h = ρgh , so (E(X), ρ1G )
is an ambit. We can also give E(X) a compact left-topological semigroup structure; rather
than a universal property, it is the fact that members of E(X) are functions that allows us
to do this. Indeed, the product is given by composition, which with our notation means that
for f1 , f2 ∈ E(X), we define x · (f1 · f2 ) = (x · f1 ) · f2 . The ambit E(X) (the distinguished
point being understood) is called the enveloping semigroup of X; we will be particularly
interested in E(M (G)), the enveloping semigroup of the universal minimal flow.
It is worth pointing out that we could have avoided some of this notational awkwardness by
switching the roles of left and right throughout, i.e. working with left G-actions and compact
right-topological semigroups. The reason we work with our left-right conventions is due to
the specific groups that we will be working with, i.e. automorphism groups of countable
first-order structures. We will point out later how a left-right switch could be made. Also
note that several of the references use the opposite left-right conventions, in particular [HS]
and [Ba].
Robert Ellis (see [E]) first proved the existence and uniqueness of M (G), and was the first
to consider the two canonical ambits S(G) and E(M (G)). As S(G) is the greatest ambit,
there is a map of ambits ϕ : S(G) → E(M (G)) (when referring to S(G) and enveloping semigroups, we will suppress the distinguished point unless there is possible confusion). He posed
the following very natural question: is ϕ : S(G) → E(M (G)) an isomorphism? Vladimir
Pestov (see [P]) observed that the existence of extremely amenable groups, groups where
M (G) is a singleton, provides a negative answer to Ellis’s question. Pestov also constructed
many other examples of groups G where S(G) and E(M (G)) were not isomorphic. The
diversity of counterexamples to Ellis’s question led Pestov to make the following conjecture.
Conjecture 1.1 (Pestov). Let G be a topological group. Then the canonical map ϕ : S(G) →
E(M (G)) is an isomorphism iff G is precompact.
Here, G is said to be precompact if the completion of its left uniformity is compact. If this
is the case, then all of S(G), M (G), and E(M (G)) are isomorphic to the left completion.
Aside from the initial work of Pestov, very little work has been done on Conjecture 1.1.
Glasner and Weiss (see [GW1]) have shown that S(Z) and E(M (Z)) are not isomorphic.
Their proof is rather difficult and uses some deep results in ergodic theory (see [F]); as such,
their methods are unlikely to generalize even to countable amenable groups.
In this paper, we address Conjecture 1.1 for groups of the form G = Aut(K) where K is
a countable first-order structure. We endow G with the topology of pointwise convergence,
turning G into a Polish group. In a mild abuse of terminology, we will call groups of this
form automorphism groups. When K is a countable set with no additional structure, we
have Aut(K) = S∞ , the group of all permutations of a countable set. More generally, automorphism groups are exactly the closed subgroups of S∞ . The work of Kechris, Pestov, and
Todorcevic [KPT] provides explicit computations of M (G) for many automorphism groups.

SAMUEL COMPACTIFICATIONS

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Having an explicit representation of M (G) aids in analyzing the properties of E(M (G)).
Along with an explicit representation of S(G) for automorphism groups (see [Z]), this allows us to address Conjecture 1.1 for some of these groups. Our first main theorem is the
following.
Theorem 1.2. Let K be any of the following:
• a countable set without structure,
• the random Kn -free graph,
• the random r-uniform hypergraph.
Then for G = Aut(K), we have S(G) ∼
6= E(M (G)).
We then turn to finding the extent to which S(G) and E(M (G)) differ. Any minimal
subflow M ⊆ S(G) is isomorphic to M (G), and it turns out that S(G) admits a retraction
onto M , i.e. a G-map ϕ : S(G) → M with ϕ|M the identity. Pestov has shown (see [P1]) that
S(G) ∼
= E(M (G)) iff the retractions of S(G) onto a minimal subflow M ⊆ S(G) separate
the points of S(G). So if S(G) ∼
6= E(M (G)), it makes sense to ask which pairs of points
cannot be separated; this will not depend on the choice of minimal subflow M ⊆ S(G).
Given x, y ∈ S(G), we say they can be separated by retractions if there is a retraction
ϕ : S(G) → M with ϕ(x) 6= ϕ(y).
Every compact left-topological semigroup S admits a smallest two-sided ideal, denoted
K(S). Our second main theorem is the following.
Theorem 1.3. There are x 6= y ∈ K(S(S∞ )) which cannot be separated by retractions.
On the way to proving Theorem 1.3, we prove some theorems of independent interest
both for general topological groups G and for S∞ . By a well-known theorem of Ellis, every
compact left-topological semigroup S contains an idempotent, an element u ∈ S which
satisfies u · u = u (see [E]). Given Y ⊆ S, write J(Y ) for the set of idempotents in Y . Our
route to proving Theorem 1.3 involves a careful understanding of when the product of two
idempotents is or is not an idempotent.
In the case G = S∞ , we are able to find large semigroups of idempotents; this is what
allows us to prove Theorem 1.3.
Theorem 1.4. There are two minimal subflows M 6= N ⊆ S(S∞ ) so that J(M ) ∪ J(N ) is
a semigroup.
It is worth noting that any minimal subflow M ⊆ S(G) is a compact subsemigroup of
S(G), so J(M ) 6= ∅.
There are some cases when it is clear that K(S(G)) contains sufficiently large semigroups
of idempotents. Given a G-flow X, recall that a pair of points x, y ∈ X is called proximal
if there is p ∈ E(X) with xp = yp; the pair (x, y) is called distal if it is not proximal. A
G-flow X is proximal if every pair from X is proximal, and X is called distal if every pair
x 6= y ∈ X is distal. If M (G) is proximal, then whenever M ⊆ S(G) is a minimal subflow, we
have J(M ) = M . If M (G) is distal and M ⊆ S(G) is a minimal subflow, then J(M ) = {u},
a single idempotent. So long as S(G) contains at least two minimal right ideals, which is
always the case when G is Polish (see [Ba]), then E(M (G)) ∼
6= S(G) in these cases. We will
discuss examples of groups with M (G) proximal and provide a partial characterization of
Polish groups G with M (G) distal.

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Theorem 1.5. Let G be a Polish group. The following are equivalent.
(1) M (G) is distal and metrizable.
i

ϕ

(2) There is a short exact sequence of groups 1 → H →
− G−
→ K → 1 with H extremely
amenable and K compact metrizable.
Furthermore, in item (2), K ∼
= M (G), and the G-action is given by k · g = kϕ(g).

2. Countable first-order structures and the Samuel compactification
In this section, we provide the necessary background on countable structures and provide
an explicit construction of the Samuel compactification of an automorphism group. The
presentation here is largely taken from [Z1].
Recall that S∞ is the group of all permutations of ω := {0, 1, 2, ...}. We can endow S∞
with the topology of pointwise convergence; a typical basic open neighborhood of the identity
is {g ∈ S∞ : g(k) = k for every k < n} for some n < ω. Notice that each of these basic open
neighborhoods is in fact a clopen subgroup.
Fix now G a closed subgroup of S∞ . A convenient way to describe the G-orbits of finite
tuples from ω is given by the notions of a Fraı̈ssé class and structure. A relational language
L = {Ri : i ∈ I} is a collection of relation symbols. Each relation symbol Ri has an arity
ni ∈ N. An L-structure A = hA, RiA i consists of a set A and relations RiA ⊆ Ani ; we say
that A is an L-structure on A. If A, B are L-structures, then g : A → B is an embedding
if g is a map from A to B such that RiA (x1 , ..., xni ) ⇔ RiB (g(x1 ), ..., g(xni )) for all relations.
We write Emb(A, B) for the set of embeddings from A to B. We say that B embeds A
and write A ≤ B if Emb(A, B) 6= ∅. An isomorphism is a bijective embedding, and an
automorphism is an isomorphism between a structure and itself. If A ⊆ B, then we say that
A is a substructure of B, written A ⊆ B, if the inclusion map is an embedding. A is finite,
countable, etc. if A is.
Definition 2.1. Let L be a relational language. A Fraı̈ssé class K is a class of L-structures
with the following four properties.
(1) K contains only finite structures, contains structures of arbitrarily large finite cardinality, and is closed under isomorphism.
(2) K has the Hereditary Property (HP): if B ∈ K and A ⊆ B, then A ∈ K.
(3) K has the Joint Embedding Property (JEP): if A, B ∈ K, then there is C which
embeds both A and B.
(4) K has the Amalgamation Property (AP): if A, B, C ∈ K and f : A → B and
g : A → C are embeddings, there is D ∈ K and embeddings r : B → D and
s : C → D with r ◦ f = s ◦ g.
If K is a countably infinite L-structure (which we will typically assume has underlying set
ω), we write Age(K) for the class of finite L-structures which embed into K. The following
is the major fact about Fraı̈ssé classes.

SAMUEL COMPACTIFICATIONS

5

Fact 2.2. If K is a Fraı̈ssé class, there is up to isomorphism a unique countably infinte
L-structure K with Age(K) = K satisfying one of the following two equivalent conditions.
(1) K is ultrahomogeneous: if f : A → B is an isomorphism between finite substructures
of K, then there is an automorphism of K extending f .
(2) K satisfies the Extension Property: if B ∈ K, A ⊆ B, and f : A → K is an
embedding, there is an embedding h : B → K extending f .
Conversely, if K is a countably infinite L-structure satisfying 1 or 2, then Age(K) is a
Fraı̈ssé class.
Given a Fraı̈ssé class K, we write Flim(K), the Fraı̈ssé limit of K, for the unique structure
K as above. We say that K is a Fraı̈ssé structure if K ∼
= Flim(K) for some Fraı̈ssé class.
Our interest in Fraı̈ssé structures stems from the following result.
Fact 2.3. For any Fraı̈ssé structure K, Aut(K) is isomorphic to a closed subgroup of S∞ .
Conversely, any closed subgroup of S∞ is isomorphic to Aut(K) for some Fraı̈ssé structure
K.
Fix S
a Fraı̈ssé class K with Fraı̈ssé limit K. Set G = Aut(K). We also fix an exhaustion
K = n AnS
, with each An ∈ K, |An | = n, and Am ⊆ An for m ≤ n. Whenever we
write K = n An , it will be assumed that the right side is an exhaustion of K. Write
Hn = {gGn : g ∈ G}, where Gn = G ∩ NAn is the pointwise stabilizer of An . We can
identify Hn with Emb(An , K),Sthe set of embeddings of An into K. Note that under this
identification, we have Hn = N ≥n Emb(An , AN ). For g ∈ G, we often write g|n for gGn ,
and we write in for Gn . The group G acts on Hn on the left; if x ∈ Hn and g ∈ G, we have
g · x = g ◦ x. For m ≤ n, we let inm ∈ Emb(Am , An ) be the inclusion embedding.
Each f ∈ Emb(Am , An ) gives rise to a dual map fˆ : Hn → Hm given by fˆ(x) = x◦f . Note
that we must specify the range of f for the dual map to make sense, but this will usually be
clear from context.
Proposition 2.4.
(1) For f ∈ Emb(Am , An ), the dual map fˆ : Hn → Hm is surjective.
(2) For every f ∈ Emb(Am , An ), there is N ≥ n and h ∈ Emb(An , AN ) with h ◦ f = iN
m.
Proof. Item 1 is an immediate consequence of the extension property. For item 2, use ultrahomogeneity to find g ∈ G with g ◦ f = im . Let N ≥ n be large enough so that ran(g|n ) ⊆ AN ,
and set h = g|n .

We now proceed with an explicit construction of S(G). First, if X is a discrete space, we
let βX be the space of ultrafilters on X. We topologize βX by declaring a typical basic open
neighborhood to be of the form {p ∈ βX : A ∈ p}, where A ⊆ X. We view X as a subset of
βX by identifying x ∈ X with the ultrafilter {A ⊆ X : x ∈ A}. If Y is a compact Hausdorff
space and ϕ : X → Y is any map, there is a unique continuous extension ϕ̃ : βX → Y .
Now let f ∈ Emb(Am , An ). The dual map fˆ extends to a continuous map f˜ : βHn → βHm .
If p ∈ βHn and f ∈ Emb(Am , An ), we will sometimes write p · f for f˜(p). Form the inverse

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D. BARTOŠOVÁ AND A. ZUCKER

limit lim βHn along the maps ı̃nm . We can identify G with a dense subspace of lim βHn by
←−
←−
associating to each g ∈ G the sequence of ultrafilters principal on g|n . The space lim βHn
←−
turns out to be the Samuel compactification S(G) (see Corollary 3.3 in [P]).
To see that S(G) is the greatest ambit, we need to exhibit a right G-action on S(G).
This might seem unnatural at first; after all, the left G-action on each Hn extends to a left
G-action on βHn , giving us a left G-action on S(G). The problem is that the left action is
not continuous when G is given its Polish topology. The right action we describe doesn’t
“live” on any one level of the inverse limit lim βHn ; we need to understand how the various
←−
levels interact.
Let πn : lim βHn → βHn be the projection map. We often write α(n) := πn (α). For
←−
α ∈ lim βHn , g ∈ G, m ∈ N, and S ⊆ Hm , we have
←−
S ∈ αg(m) ⇔ {x ∈ Hn : x ◦ g|m ∈ S} ∈ α(n)
where n ≥ m is large enough so that ran(g|m ) ⊆ An . Notice that if g|m = h|m = f ,
then αg(m) = αh(m) := α · f := λαm (f ). By distinguishing the point 1 ∈ lim βHn with
←−
1(m) principal on im , we endow S(G) with the structure of a G-ambit, and (S(G), 1) is the
greatest ambit (see Theorem 6.3 in [Z]).
Using the universal property of the greatest ambit, we can define a left-topological semigroup structure on S(G): Given α and γ in lim βHn , m ∈ N, and S ⊆ Hm , we have
←−
S ∈ αγ(m) ⇔ {f ∈ Hm : S ∈ α · f } ∈ γ(m).
If α ∈ S(G) and S ⊆ Hm , a useful shorthand is to put
α−1 (S) = {f ∈ Hm : S ∈ α · f }.
Then the semigroup multiplication can be written as
S ∈ αγ(m) ⇔ α−1 (S) ∈ γ(m).
Notice that for fixed α, αγ(m) depends only on γ(m); indeed, if α ∈ lim βHn , p ∈ βHm , and
←−
S ⊆ Hm , we have S ∈ α · p iff α−1 (S) ∈ p. In fact, α · p = λ̃αm (p), where the map λ̃αm is the
continuous extention of λαm to βHm .
As promised in the introduction, we now explain the reason behind our left-right conventions. The primary reason behind considering right G-flows is because for G = Aut(K), the
left uniformity is very natural to describe. Namely, every entourage contains an entourage
of the form {(g, h) ∈ G × G : g|m = h|m }. This leads naturally to considering the embeddings from Am to K. If we wanted to consider the right uniformity, we would instead be
considering partial isomorphisms of K with range Am , which are less easily described.
3. KPT correspondence
In this section, we provide a brief review of KPT correspondence. For proofs of the results
in this section, see [KPT], [NVT], or [Z].
Let L be a relational language and L∗ = L ∪ S, where S = {Si : i ∈ N} and the Si
are new relational symbols of arity ni . If A is an L∗ -structure, write A|L for the structure

SAMUEL COMPACTIFICATIONS

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obtained by throwing away the interpretations of the relational symbols in L∗ \ L. If K∗
is a class of L∗ -structures, set K∗ |L = {A∗ |L : A∗ ∈ K∗ }. If K = K∗ |L and K∗ is closed
under isomorphism, we say that K∗ is an expansion of K. If A∗ ∈ K∗ and A∗ |L = A, then
we say that A∗ is an expansion of A, and we write K∗ (A) for the set of expansions of A
in K∗ . If f ∈ Emb(A, B) and B∗ ∈ K∗ (B), we let B∗ · f be the unique expansion of A
so that f ∈ Emb(B∗ · f, B∗ ). The expansion K∗ is precompact if for each A ∈ K, the set
{A∗ ∈ K∗ : A∗ |L = A} is finite.
If K∗ is an expansion of the Fraı̈ssé class K, we say that the pair (K∗ , K) is reasonable if for
any A, B ∈ K, embedding f : A → B, and expansion A∗ of A, then there is an expansion
B∗ of B with f : A∗ → B∗ an embedding. When K∗ is also a Fraı̈ssé class, we have the
following equivalent definition.
Proposition 3.1. Let K∗ be a Fraı̈ssé expansion class of the Fraı̈ssé class K with Fraı̈ssé
limits K∗ , K respectively. Then the pair (K∗ , K) is reasonable iff K∗ |L ∼
= K.
Set Fin(K) = {A ∈ K : A ⊆ K}. Suppose (K∗ , K) is reasonable and precompact. Set
~ : hA, S|
~ A i ∈ K∗ whenever A ∈ Fin(K)}.
XK∗ := {hK, Si
We topologize this space by declaring the basic open neighborhoods to be of the form
N (A∗ ) := {K0 ∈ XK∗ : A∗ ⊆ K0 }, where A∗ is an expansion of some A ∈ Fin(K). We
can view XK∗ as a closed subspace of
Y
{A∗ : A∗ ∈ K∗ (A)}.
A∈Fin(K)

Notice that since (K∗ , K) is precompact, XK∗ is compact. If
compatible metric is given by

S

n

An = K is an exhaustion, a

~ hK, T~ i = 1/k(S,
~ T~ ),
d(hK, Si,
~ T~ ) is the largest k for which hAk , S|
~A i∼
where k(S,
= hAk , T~ |Ak i.
k
We can now form the (right) logic action of G = Aut(K) on XK∗ by setting K0 · g to be
the structure where for each relation symbol S ∈ S, we have
0

0

S (K ·g) (x1 , ..., xn ) ⇔ S K (g(x1 ), ..., g(xn )).
This action is jointly continuous, turning XK∗ into a G-flow. For readers used to left logic
actions, acting on the right by g is the same as acting on the left by g −1 .
First let us consider when XK∗ is a minimal G-flow.
Definition 3.2. We say that the pair (K∗ , K) has the Expansion Property (ExpP) when for
any A∗ ∈ K∗ , there is B ∈ K such that for any expansion B∗ of B, there is an embedding
f : A∗ → B ∗ .
Proposition 3.3. Let K∗ be a reasonable, precompact Fraı̈ssé expansion class of the Fraı̈ssé
class K with Fraı̈ssé limits K∗ , K respectively. Let G = Aut(K). Then the G-flow XK∗ is
minimal iff the pair (K∗ , K) has the ExpP.

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D. BARTOŠOVÁ AND A. ZUCKER

Expansion classes are particularly inetesting when K∗ has the following combinatorial
property.
Definition 3.4. Let C be a class of finite structures.
(1) We say that A ∈ C is a Ramsey object if for any r ≥ 2 and any B ∈ C with A ≤ B,
there is C ∈ C with B ≤ C so that for any coloring c : Emb(A, C) → r, there is
h ∈ Emb(B, C) with |c(h ◦ Emb(A, B))| = 1.
(2) We say that C has the Ramsey Property (RP) if every A ∈ C is a Ramsey object.
The following is one of the major theorems in [KPT]. This theorem in its full generality
is proven in [NVT].
Theorem 3.5. Let K∗ be a reasonable, precompact Fraı̈ssé expansion class of the Fraı̈ssé
class K with Fraı̈ssé limits K∗ , K, respectively. Let G = Aut(K). Then XK∗ ∼
= M (G) iff the
pair (K∗ , K) has the ExpP and K∗ has the RP.
Pairs (K∗ , K) of Fraı̈ssé classes which are reasonable, precompact, satisfy the ExpP, and
where K∗ has the RP are called excellent. In particular, if K = Flim(K), G = Aut(K), and
there is an expansion class K∗ so that (K∗ , K) is excellent, then M (G) is metrizable. The
following converse is one of the major theorems of [Z].
Theorem 3.6. Let K be a Fraı̈ssé class with K = Flim(K) and G = Aut(K). If M (G) is
metrizable, then there is an expansion class K∗ so that (K∗ , K) is excellent.

4. Ellis’s problem for random relational structures
In S
this section, we prove Theorem 1.2. Let G = Aut(K) for some Fraı̈ssé structure
K = n An . If T ⊆ Hm , n ≥ m, and f ∈ Emb(Am , An ), set
f (T ) = {s ∈ Hn : s ◦ inm ∈ T }.
This is a minor abuse of notation, since f also denotes a map from Am to An ; however, this
map induces a continuous embedding of the Boolean algebra P(Hm ) into P(Hn ).
We will freely identify P(Hm ) with 2Hm ; in particular, G acts on 2Hm by right shift, where
for ϕ ∈ 2Hm , f ∈ Hm , and g ∈ G, we have ϕ · g(f ) = ϕ(g · f ).
Definition 4.1. We call a subset S ⊆ Hm minimal if the flow χS · G ⊆ 2Hm is minimal.
The formulation of Ellis’s problem we will work with is the one concerning retractions
given by Pestov. We will be interested in whether every pair x 6= y ∈ S(G) can be separated
by retractions. A characterization of when this occurs for discrete groups can be found in
[Ba] (see Proposition 11). We first prove a similar characterization for automorphism groups
in the next two lemmas.

SAMUEL COMPACTIFICATIONS

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Before proceeding, a quick remark on notation is in order. If X is a G-flow, then there
is a unique map of ambits ϕ : S(G) → E(X). If x ∈ X and p ∈ S(G), we write x · p for
x · ϕ(p).
Lemma 4.2. Suppose α, γ ∈ S(G) cannot be separated by retractions, and let S ⊆ Hm be
minimal. Then S ∈ α(m) ⇔ S ∈ γ(m).
Proof. Let M ⊆ S(G) be a minimal subflow, and consider the non-empty closed subsemigroup {p ∈ M : χS · p = χS }. By Ellis’s theorem, let u ∈ M be an idempotent with
χS · u = χS . As the left multiplication λu : S(G) → M is a retraction, we must have
u · α = u · γ. Therefore χS · α = χS · γ, and α−1 (S) = γ −1 (S) := T . It follows that im ∈ T
iff S ∈ α(m) iff S ∈ γ(m).

For each m < ω, let Bm ⊆ P(Hm ) be the Boolean algebra generated by the minimal
0
subsets of Hm . Let Bm
be the Boolean algebra {T ⊆ Hm : ∃n ≥ m(inm (T ) ∈ Bn )}.
Lemma 4.3. Fix M ⊆ S(G) a minimal. The following are equivalent.
(1) Retractions of S(G) onto M separate points of S(G),
0
= P(Hm ).
(2) For every m < ω, we have Bm

Proof. Suppose that there are α 6= γ ∈ S(G) which cannot be separated by retractions. Find
m < ω with α(m) 6= γ(m), and find T ⊆ Hm with T ∈ α(m), T 6∈ γ(m). Note that for
every n ≥ m, we have inm (T ) ∈ α(n) and inm (T ) 6∈ γ(n). Towards a contradiction, suppose
for some n ≥ m that inm (T ) was a Boolean combination of minimal set A1 , ..., Ak ⊆ Hn .
By our assumption, α(n) and γ(n) agree on the membership of each Ai , hence also on the
membership of T , a contradiction.
0
0
) denote
. Let S(Bm
Conversely, suppose that T ⊆ Hm is not in the Boolean algebra Bm
0
0
0
the Stone space of Bm ; form the inverse limit lim S(Bn ), and let p ∈ lim S(Bn ) be chosen
←−
←−
so that p(m) doesn’t decide T . Then find α, γ ∈ lim βHn with T ∈ α(m), T 6∈ γ(m) which
←−
both extend p. Then α and γ cannot be separated by retractions.

Notice that item (2) of Lemma 4.3 does not depend on M . In general, the relation of
whether x 6= y ∈ S(G) can be separated by retractions does not depend on the minimal
subflow of S(G) chosen, but we postpone this discussion until the end of section 6 (see the
discussion after Theorem 5.11).
Now suppose that (K∗ , K) is an excellent pair of Fraı̈ssé classes. Given a set of expansions
E ⊆ K∗ (Am ) and K0 ∈ XK∗ , let Hm (E, K0 ) = {f ∈ Hm : Am (f, K0 ) ∈ E}. If n ≥ m and
A0n ∈ K∗ (An ), we set Emb(E, A0n ) = {f ∈ Emb(Am , An ) : Am (f, A0n ) ∈ E}.
Proposition 4.4. S ⊆ Hm is minimal iff there is K0 ∈ XK∗ and E ⊆ K∗ (Am ) so that
S = Hm (E, K0 )
Proof. One direction is easy once we note that given E ⊆ K∗ (Am ), the map ϕE : XK∗ → 2Hm
given by ϕE (K0 ) = Hm (E, K0 ) is a map of G-flows. In the other direction, let S ⊆ Hm be
minimal. Then Y := χS · G ⊆ 2Hm is a minimal G-flow, so fix a G-map ϕ : XK∗ → Y . Note
that ϕ(K∗ ) must be a G∗ -fixed point, so by ultrahomogeneity of K∗ it must be of the form

10

D. BARTOŠOVÁ AND A. ZUCKER

Hm (E, K∗ ) for some E ⊆ K∗ (Am ). It follows that ϕ = ϕE , so in particular S = Hm (E, K0 )
for some K0 ∈ XK∗ .

The main tool allowing us to prove Theorem 1.2 is an explicit characterization of M (G)
for certain autormorphism groups G. The following facts can be found in [KPT].
Fact 4.5. Let K = Age(K), where K is any of the structures in the statement of Theorem
1.2. Let K∗ be the class of linearly ordered members of K. Then (K∗ , K) is an excellent pair.
Setting G = Aut(K), then M (G) = XK∗ is the space of linear orders of K.
The next theorem is the simplest case of Theorem 1.2. The following notion will be useful
in the proof. Given T ⊆ Hm and N ≥ m, an N -pattern of T is a set S ⊆ Emb(Am , AN ) so
that there is y ∈ HN with S = {f ∈ Emb(Am , AN ) : y ◦ √
f ∈ T }. We Will be using Stirling’s
formula which states that asymptotically n! is equal to 2πn( ne )n .
Theorem 4.6. Let K = {xn : n < ω} be a countable set with no structure (so G ∼
= S∞ ),
0
⊆ 2Hm is meager. In particular, any
and set Am = {xi : i < m}. Then for every m ≥ 2, Bm
0
.
T ∈ Hm whose orbit is dense is not in Bm
Proof. Let T ⊆ Hm have dense orbit. So for every N ≥ m, every S ⊆ Emb(Am , AN ) is an
N -pattern of T . Towards a contradiction, suppose for some n ≥ m that inm (T ) was a Boolean
combination of minimal sets B1 , ..., Bk ⊆ Hn . Let N >> n; we will obtain a contradiction
N
m
by counting the number of N -patterns in inm (T ), which by assumption is 2m!(m) ≥ 2N /2 .
Since |K∗ (An )| = n! and since there are N ! linear orders on AN , this gives us 2n! N !
possible N -patterns for each Bi by Proposition 4.4. Therefore any N -pattern of T must be
a Boolean combination of some k of these N -patterns. Each choice of k patterns results
k
in at most 22 Boolean combinations, so the total number of possible patterns is at most
k
22 (2n! N !)k patterns. Noting that n and k remain fixed as we let N grow large, we have
that asymptotically there are CN · ekN possible N -patterns of inm (T ), which is far less than
m
2N /2 , a contradiction.

We now consider the case where K = Flim(K) is the random r-uniform hypergraph for
some r ≥ 2. In order to generalize the arguments
in the proof of Theorem 4.6, we will
S
need some control over the exhaustion K = n An . We will do this by not specifying an
exhaustion in advance, but instead determining parts of it as we proceed.
We will need the following notion. With K as above, let A ⊆ B ∈ K, and let C ⊆ D ∈ K.
We say that D extends C along A ⊆ B if for any f ∈ Emb(A, C), there is an h ∈ Emb(B, D)
with h|A = f .
Given C ∈ K, write |C| for the number of vertices in C.
Lemma 4.7. Let K be the class of r-uniform hypergraphs for some r ≥ 2. Let e ⊆ B ∈ K,
where e ∈ K is the hypergraph on r vertices consisting of an edge, and let C ∈ K with
|C| = N . Then there is D ∈ K extending C along e ⊆ B with |D| ≤ cN r−1 for some
constant c depending only on |B|.

SAMUEL COMPACTIFICATIONS

11

Proof. Recall that given an r-uniform hypergraph C, a matching is a subset of the edges
of C so that each vertex is included in at most one edge. By Baranyai’s theorem [B],
the edge set
into M1 , ..., M` with each Mi a matching and with
 of C can be partitioned
rdN/re
r−1
`≤
/dN/re) ≈ c0 N . For each i ≤ ` and j ≤ r!, let Dij be a set of |B| − r new
r
S S
vertices. We will define the hypergraph D on vertex set C ∪ i≤` j≤r! Dij . First add edges
to D so that Dij ∼
= B \ e. For each e0 ∈ Mi , enumerate the embeddings fj : e → C with
0
range e . Add edges to D so that each fj extends to an embedding hj : B → D with range
e0 ∪ Dij . This is possible since each Mi is a matching. The hypergraph D has |D| ≤ cN r−1
as desired.

Theorem 4.8. Let K be the class of r-uniform hypergraphs for r ≥ 2, with K = Flim(K).
Let Ar ⊆ K be an edge on r vertices. Then if T ⊆ Hr has dense orbit, then T 6∈ Br0 .
Remark. Though we are not specifying an exhaustion in advance, we will still use some of the
associated notation. In particular, when we write Am for some m < ω, we mean a subgraph
of K on m vertices.
Proof. Suppose towards a contradiction that there were some graph An ⊇ Ar so that inr (T )
was a Boolean combination of minimal sets B1 , ..., Bk ⊆ Hn . Let N  n, and fix a graph
AN ⊇ An with at least Nr /2 edges. Let AN 0 ⊇ AN extend AN along Ar ⊆ An with
N 0 ≈ cN r−1 as guaranteed by Lemma 4.7. We will obtain a contradiction by counting the
number of N 0 -patterns of inr (T ). Exactly as in the proof of Theorem 4.6, there are at most
k
0
r−1
22 (2n! N 0 !)k ≈ CN 0 · ekN ≈ CN r−1 · ekN
many N 0 -patterns. But since T has dense orbit,
N
r

there must be at least 2( r )/2 ≈ 2cN N 0 -patterns of inr (T ), a contradiction.
We next turn to the class K of Kr -free graphs for some r ≥ 3. We will need a result similar
to Lemma 4.7, but the given proof will not work as the construction doesn’t preserve being
Kr -free. Recall that the Ramsey number R(r, n) of r and n is the smallest integer such that
any graph on R(r, n) vertices either contains a clique of size r or an independent set of size
n.
Lemma 4.9. Let K be the class of Kr -free graphs for some r ≥ 3. Let e ⊆ B ∈ K, where
e is an edge, and let C ∈ K with |C| = N . Then there is D ∈ K extending C along e ⊆ B
with |D| ≤ cN 2(r−1)/(r−2) .
Proof. Let R(r, n) be the Ramsey number of r and n. In [AKS], it is shown that R(r, n) =
o(nr−1 ). Since C is Kr -free, this implies that C has an independent set of size at least
N 1/(r−1) ; by repeatedly removing independent sets, we see that the chromatic number of C
is at most ` ≈ N (r−2)/(r−1) . Write C = C1 t · · · t C` so that each Ci is an independent set.
For every ordered pair (i, j) of distinct indices with i, j ≤S`, let D(i,j) be a set of |B| − 2 new
vertices. We will define the graph D on vertex set C ∪ {i,j}∈[`]2 (D(i,j) ∪ D(j,i) ). First add
edges to D so that D(i,j) ∼
= B \ e; fix h0 : B \ e → D(i,j) an isomorphism. Write e = {a, b}; if
f : e → C with f (a) = i and f (b) = j, then add edges to D so that h0 ∪ f := h : B → D is an
embedding with range f (e)∪D(i,j) . The hypergraph D is Kr -free and has |D| ≤ cN 2(r−2)/(r−1)
as desired.


12

D. BARTOŠOVÁ AND A. ZUCKER

Theorem 4.10. Let K be the class of Kr -free graphs for some r ≥ 3, with K = Flim(K).
Let A2 ⊆ K be an edge. Then if T ⊆ H2 has dense orbit, then T 6∈ B20 .
As in the proof of Theorem 4.8, we will not specify an exhaustion in advance, but we will
still use some of the notational conventions.
Proof. Suppose towards a contradiction that there were some graph An ⊇ A2 so that in2 (T )
was a Boolean combination of minimal sets B1 , ..., Bk ⊆ Hn . Let N  n, and fix a graph
AN ⊇ An with at least n2 /r edges. Let AN 0 ⊇ AN extend AN along A2 ⊆ An with
N 0 ≈ cN 2(r−2)/(r−1) as guaranteed by Lemma 4.9. We now obtain a contradiction by counting
0
2(r−1)/(r−2)
N 0 -patterns. Once again, there are at most CN 0 · ekN ≈ CN 2(r−2)/(r−1) · ekN
many
N
/r
0
n
cN 2
(
)
2
N -patterns in i2 (T ), which contradicts the fact that there are at least 2
≈2
many
N 0 -patterns.

We end this section with a conjecture. While it is a strict sub-conjecture of Conjecture
1.1, we think it might be more easily approached.
Conjecture 4.11. Let G be a closed, non-compact subgroup of S∞ with metrizable universal
minimal flow. Then S(G) 6∼
= E(M (G)).

5. A closer look at S∞
In this section, we take a closer look at S(S∞ ), with an eye towards understanding which
pairs of points x 6= y ∈ S(S∞ ) can be separated by retractions. We view S∞ as the group of
permutations of ω. We can realize ω as a Fraı̈ssé structure in the empty language. We set
An = n, so that Hn is the set of all injections from n into ω, and for m ≤ n, Emb(Am , An )
is the set of all injections from m into n. We will often abuse notation and write s ∈ Hm as
the tuple (s0 , ..., sm−1 ), where si = s(i).
We start by developing some notions for any automorphism group. Let f ∈ Emb(Am , An ).
If F ⊆ P(Hm ) is a filter, then we write f (F) for the filter generated by {f (T ) : T ∈ F}.
If H ⊆ P(Hn ) is a filter, then f˜(H) is the push-forward filter {T ⊆ Hm : f (T ) ∈ H}. This
may seem like a conflict of notation since f˜ : βHn → βHm is the extended dual map of f .
We canTjustify this notation as follows. To each filter H on Hn , we associate the closed set
XH := A∈H A ⊆ βHn . Conversely, given a closed set X ⊆ βHn , we can form the filter of
clopen neighborhoods FX := {A ⊆ Hn : X ⊆ A}. Then we obtain the identity
Xf˜(H) = f˜(XH ).
A similar identity holds given a filter F on Hm :
Xf (F ) = f˜−1 (XF ).
Let Y ⊆ S(G) be closed. Let πm : S(G) → βHm be the projection map. Then πm (Y )
Y
is a closed subset of βHm . Write Fm
for Fπm (Y ) . For n ≥ m, the filter FnY extends the
n
Y
n
Y
Y
filter im (Fm ) and ı̃m (Fn ) = Fm . Conversely, given filters Fm on Hm for every m < ω such

SAMUEL COMPACTIFICATIONS

13

that Fn extends inm (Fm ) and with ı̃nm (Fn ) = Fm , there is a unique closed Y ⊆ S(G) with
Y
Fm = Fm
for each m < ω. We will call such a sequence of filters compatible.
M
We will need to understand the filters Fm
when M ⊆ S(G) is a minimal subflow. It turns
out that these filters are characterized by a certain property of their members.
Definition 5.1. Given T ⊆ Hm , we say that T is thick if either of the following equivalent
items hold (see [Z1]).
(1) T ⊆ Hm is thick iff χHm ∈ χT · G.
(2) T ⊆ Hm is thick iff for every n ≥ m, there is s ∈ Hn with s ◦ Emb(Am , An ) ⊆ T .
We can now state the following fact from [Z1].
Theorem 5.2. Let G be an automorphism group, and let M ⊆ S(G) be closed. Then M is
M
a minimal subflow iff each Fm
is a maximal filter of thick sets.
Another observation is the following.
Y
and f ∈ Emb(Am , An ).
Proposition 5.3. Say Y ⊆ S(G) is a subflow, and let T ∈ Fm
Y
Then f (T ) ∈ Fn .

Proof. Pick g ∈ G with g|m = f . Then for any α ∈ S(G), we have T ∈ αg(m) iff f (T ) ∈ α(n).
As Y is G-invariant, the result follows.

We now turn our attention to G = S∞ . Let {σi : i < m!} T
list the permutations of m, i.e.
M
. Call S ⊆ Hm
the members of Emb(Am , Am ). Then by Proposition 5.3, i σi (T ) ∈ Fm
saturated if whenever (a0 , ..., am−1 ) ∈ S and σ is a permutation, then (aσ(0) , ..., aσ(m−1) ) ∈ S.
M
has a base of saturated sets.
We have just shown that Fm
Let us recall Ramsey’s theorem: for every k, `, r < ω with k ≤ `, there is m ≥ ` so that
for every coloring c : [m]k → r, there is A ⊆ m with |A| = ` and |c([A]k )| = 1. Ramsey’s
theorem will
consequences for the collection of thick subsets of Hm .
F
F have interesting
n
Let ϕ : n Hn → n [ω] be the order forgetful map, i.e. for (y0 , ..., ym−1 ) ∈ Hm , we set
ϕ(y0 , ..., ym−1 ) = {y0 , ..., ym−1 } ∈ [ω]m . Any filter F on Hm pushes forward to a filter ϕ(F)
on [ω]m . We can define a thick subset of [ω]m in a very similar fashion to a thick subset of
Hm ; more precisely, we say T ⊆ [ω]m is thick iff for every n ≥ m, there is s ∈ [ω]n with
[s]m ⊆ T . Call S ⊆ [ω]m thin if it is not thick. We now have the following crucial corollary
of Ramsey’s theorem: if T ⊆ [ω]m is thick and T = T0 ∪ · · · ∪ Tk , then some Ti is thick. In
particular, if H is a thick filter on [X]m , i.e. a filter containing only thick sets, then we can
extend H to a thick ultrafilter. It also follows that for every m < ω, the collection of thin
subsets of [ω]m forms an ideal.
M
Theorem 5.4. Let M ⊆ S(S∞ ) be a minimal right ideal. Then for every m < ω, ϕ(Fm
)
m
−1
is a thick ultrafilter. Conversely, if p ∈ β[ω] is a thick ultrafilter, then {ϕ (T ) : T ∈ p}
generates a maximal thick filter on Hm , hence there is a minimal M ⊆ S(S∞ ) with p =
M
ϕ(Fm
).

14

D. BARTOŠOVÁ AND A. ZUCKER

M
) is a thick filter. Towards a contradiction, suppose it is not an ultrafilProof. Clearly ϕ(Fm
M
M
ter, and extend it to a thick ultrafilter p ∈ β[ω]m . Let T ∈ p \ ϕ(Fm
). Then ϕ−1 (T ) 6∈ Fm
.
−1
M
However, ϕ (T ) ∩ S is thick for every saturated S ∈ Fm . As saturated sets form a base for
M
M
.
, this contradicts the maximality of Fm
Fm
m
Now let p ∈ β[ω] be a thick ultrafilter. Then F := {ϕ−1 (T ) : T ∈ p} generates a thick
filter. Suppose S ⊆ Hm and {S} ∪ F generated a thick filter strictly larger than F. We may
assume S is saturated. Then ϕ(S) ∈ p, so ϕ−1 (ϕ(S)) = S ∈ F, a contradiction.


Notice that if p ∈ β[ω]n is thick and m ≤ n, then there is a unique thick ultrafilter
q ∈ β[ω]m with the property that {a ∈ [ω]n : [a]m ⊆ S} ∈ p for every S ∈ q. Certainly
such a q must be unique. To see that this q exists, suppose [ω]m = S t T . Then the set
n
(p) for this q. If
{a ∈ [ω]n : [a]m ∩ S 6= ∅ and [a]m ∩ T 6= ∅} is not thick. We will write πm
M
n
M
M ⊆ S(G) is a minimal right ideal and p = ϕ(Fn ), then we have πm (p) = ϕ(Fm
).
Let LO(ω) be the space of linear orders on ω. Viewed as a subset of the right shift 2H2 ,
LO(ω) becomes an S∞ -flow. It is known (see [KPT] or [GW]) that LO(ω) ∼
= M (S∞ ). Indeed,
we saw in section 4 that if K is the class of finite sets and K∗ is the class of finite linear orders,
then (K∗ , K) is an excellent pair, and XK∗ ∼
= LO(ω). If M ⊆ S(S∞ ) is a minimal right ideal
and <∈ LO(ω), then the map λ : M → LO(ω) given by λ(α) = < ·α := limgi →α < ·gi is an
S∞ -flow isomorphism. We will often write <α for < ·α, and we will write > for the reverse
linear order of <.
If <0 , <1 ∈ LO(ω) and m ≥ 2, define the set
Am (<0 , <1 ) = {{a0 , ..., am−1 } ∈ [ω]m : ∀i, j < m(ai <0 aj ⇔ ai <1 aj )}
and define Bm (<0 , <1 ) = Am (<0 , >1 ). If s ∈ Hm , we say that <0 and <1 agree on s if
ϕ(s) ∈ Am (<0 , <1 ), and we say that they anti-agree on s if ϕ(s) ∈ Bm (<0 , <1 ). When
m = 2 we often omit the subscript. If M is a minimal right ideal, then ϕ(F2M ) contains
exactly one of A(<0 , <1 ) or B(<0 , <1 ). Let AM ⊆ LO(ω) × LO(ω) be defined AM = {(<0
, <1 ) : A(<0 , <1 ) ∈ ϕ(F2M )}. Then AM is certainly reflexive and symmetric. To see that AM
is an equivalence relation, note that A(<0 , <1 ) ∩ A(<1 , <2 ) ⊆ A(<0 , <2 ). Furthermore, AM
has exactly two equivalence classes; this is because B(<0 , <1 ) ∩ B(<1 , <2 ) ⊆ A(<0 , <2 ).
Lemma 5.5. Let M ⊆ S(S∞ ) be a minimal right ideal, and let (<0 , <1 ) ∈ AM . Then for
M
any m < ω, we have Am (<0 , <1 ) ∈ ϕ(Fm
).
Proof. Let {fi : i < k} enumerate Emb(A2 , Am ). Then
this is exactly the desired set.

T

i

M
fi (ϕ−1 (A(<0 , <1 ))) ∈ Fm
, and


Lemma 5.6. Let M ⊆ S(S∞ ) be a minimal right ideal, and let α ∈ M . Then the following
are equivalent:
(1) α is an idempotent,
(2) For any <∈ LO(ω), we have (<, <α ) ∈ AM ,
(3) There is <∈ LO(ω) with (<, <α ) ∈ AM .

SAMUEL COMPACTIFICATIONS

15

Proof. Suppose α ∈ M is an idempotent, and let <∈ LO(ω). Then considering i2 = (x0 , x1 ) ∈
Emb(A2 , A2 ), we have x0 <α x1 iff {f ∈ H2 : f0 < f1 } ∈ α(2). But since α is an idempotent,
this is equivalent to {f ∈ H2 : f0 <α f1 } ∈ α(2). But this implies that ϕ−1 (A(<, <α )) ∈ α(2),
implying that A(<, <α ) ∈ ϕ(F2M ).
Conversely, suppose α ∈ M and <∈ LO(ω) with (<, <α ) ∈ AM . If f ∈ Emb(A2 , An ), then
we have f0 <α f1 iff {s ∈ Hn : s(f0 ) < s(f1 )} ∈ α(n). By Lemma 5.5, we see that this is iff
{s ∈ Hn : s(f0 ) <α s(f1 )} ∈ α(n). It follows that < ·α · α =< ·α, so α is idempotent.

Theorem 5.7. Let M, N ⊆ S(S∞ ) be minimal right ideals. The following are equivalent.
(1) AM = AN ,
(2) If u ∈ M and v ∈ N are idempotents, then uv ∈ M is also idempotent.
Proof. Suppose AM 6= AN , with (<0 , <1 ) ∈ AN \ AM . Find u ∈ M with <0 ·u = <0 , and
find v ∈ N with <0 ·v = <1 . By Lemma 5.6, u and v are idempotents and uv is not an
idempotent.
Conversely, suppose u ∈ M and v ∈ N are idempotents with uv not idempotent. Find
<0 ∈ LO(ω) with <0 ·u = <0 , and let <1 = <0 ·v. Since v is idempotent, we have by Lemma
5.6 that (<0 , <1 ) ∈ AN ; but since uv is not idempotent, we have (<0 , <1 ) 6∈ AM .

It is easy to construct minimal right ideals M, N ⊆ S(S∞ ) with AM 6= AN . Let <0
m
, <1 ∈ LO(ω) be linear orders so that for every m < ω, there are sm = (sm
0 , ..., sm−1 ) ∈ Hm
m
m
m
m
and t = (t0 , ..., tm−1 ) ∈ Hm so that <0 and <1 agree on s and anti-agree on tm . Let
M ⊆ S(S∞ ) be a minimal subflow with ϕ−1 (A(<0 , <1 )) ∈ F2M , and let N ⊆ S(S∞ ) be a
minimal subflow with ϕ−1 (B(<0 , <1 )) ∈ F2N . Then (<0 , <1 ) ∈ AM \ AN .
We now turn our attention to constructing M 6= N ⊆ S(S∞ ) minimal right ideals with
AM = AN ; this will prove Theorem 1.4 as a corollary of Theorem 5.7. To this end, we will
construct two thick ultrafilters p 6= q ∈ β[ω]3 with π23 (p) = π23 (q), so that whenever M and
N are minimal subflows of S(S∞ ) with ϕ(F3M ) = p and ϕ(F3N ) = q, then ϕ(F2M ) = ϕ(F2N ).
In particular, this implies that AM = AN .
Recall that a selective ultrafilter is an ultrafilter p on ω with the property that for any
finite coloring c : [ω]2 → r, there is a p-large set A ⊆ ω which is monochromatic for c.
Another way of saying this is as follows. Given a set A ⊆ ω, set λA = [A]2 , and if F is a
filter on ω, let λF be the filter generated by {λA : A ∈ F}. Then the ultrafilter p is selective
iff λp is an ultrafilter. The existence of selective ultrafilters is independent of ZFC.
We will be considering the following generalizations of selective ultrafilters. Let m < ω.
If T ⊆ [ω]m , we set λT = {s ∈ [ω]m+1 : [s]m ⊆ T }. If n > m, we set λ(n−m) (T ) = {s ∈ [ω]n :
[s]m ⊆ T }. Notice that the λ(n−m) operation is the same as applying λ (n − m)-many times,
justifying this notation. If F is a filter on [ω]m , we let λ(n−m) F be the filter generated by
{λ(n−m) T : T ∈ F}. It can happen that for some T ∈ F we have λ(n−m) T = ∅. We will
usually be working under assumptions that prevent this from happening. For instance, if
T ⊆ [ω]m is thick, then λ(n−m) T 6= ∅ for every n > m. Even better, if F is a thick filter on
[ω]m , then λ(n−m) F is a thick filter on [ω]n .

16

D. BARTOŠOVÁ AND A. ZUCKER

Definition 5.8. Let p ∈ β[ω]m be a thick ultrafilter. We say that p is (m, n)-selective if
λ(n−m) p is an ultrafilter. We say that p is weakly (m, n)-selective if there is a unique thick
ultrafilter extending the filter λ(n−m) p.
If p ∈ β[ω]m is a thick ultrafilter and q ∈ β[ω]n is a thick ultrafilter extending λ(n−m) p,
n
then we have πm
(q) = p. Therefore to prove Theorem 1.4, it is enough to construct a
thick ultrafilter p ∈ β[ω]2 which is not weakly (2, 3)-selective. Indeed, if p ∈ β[ω]2 is not
weakly (2, 3)-selective, then there are thick ultrafilters q0 6= q1 both extending the filter λp,
so π23 (q0 ) = π23 (q1 ).
Our construction proceeds in two parts. First we define a certain type of pathological
subset of [ω]3 and show that its existence allows us to construct p ∈ β[ω]2 which is not
weakly (2, 3)-selective. Then we show the existence of such a pathological set.
We begin by developing some abstract notions. Let Y be a set, and let I be a proper
ideal on Y . Write S ⊆I T if S \ T ∈ I. Let ψ : P(Y ) → P(Y ) be a map satisfying ψ 2 = ψ,
S ⊆ ψ(S), S ⊆ T ⇒ ψ(S) ⊆ ψ(T ), and ψ(∅) = ∅. Call a set S ⊆ Y ψ-closed or just closed if
ψ(S) = S, and call S near-closed if there is a closed set T with S∆T ∈SI. Call a set S ⊆ Y
(< ℵ0 )-near-closed if there are k < ω and closed T0 , ..., Tk−1 with S∆( i<k Ti ) ∈ I. Notice
that a finite union of near-closed sets is (< ℵ0 )-near-closed.
Now suppose S ⊆ Y is a set which is not (< ℵ0 )-near-closed. If p, q ∈ βY , we say that p
ψ-intertwines q over S modulo I if the following three items all hold:
(1) {S \ T : T is near-closed and T ⊆I S} ⊆ p,
(2) {ψ(T ) \ S : T ∈ p, T ⊆ S} ⊆ q,
(3) p and q extend the filterdual of I.
If ψ, S, and I are understood, we will just say that p intertwines q. Notice in (1) that
if T is near-closed with T ⊆I S, then S ∩ T is also near-closed, so it is enough to consider
near-closed T with T ⊆ S.
Lemma 5.9. Fix S ⊆ Y which is not (< ℵ0 )-near-closed.
(1) If B ⊆ Y with B ∈ I, then B is near-closed. Hence S 6∈ I.
(2) There are p, q ∈ βY so that p intertwines q.
Proof. The first part follows since the empty set is closed.
Since S is not (< ℵ0 )-near-closed, we have that {S \ T : T near-closed and T ⊆I S}
generates a filter F extending the filterdual of I. Let p ∈ βY be any ultrafilter extending
F.
Now let T ∈ p. Then ψ(T ) \ S 6∈ I; otherwise we would have ψ(T ) ⊆I S, so S \ ψ(T ) ∈ p,
contradicting that T ∈ p. Also note by monotonicity of ψ that (ψ(T0 ) ∩ ψ(T1 )) \ S ⊇
ψ(T0 ∩ T1 ) \ S, so the collection {ψ(T ) \ S : T ∈ p} generates a filter H avoiding I; letting q
be any ultrafilter extending both H and the filterdual of I, we see that p intertwines q. 
We now apply these ideas. Let Y = [ω]3 , and let I be the thin ideal. Given T ⊆ [ω]3 , view
T as a 3-uniform hypergraph, and form the shadow graph ∂T := {{a, b} ∈ [ω]2 : ∃c({a, b, c} ∈

SAMUEL COMPACTIFICATIONS

17

T )}. Define ψ(T ) = λ∂T . In words, ψ(T ) is the largest hypergraph with ∂ψ(T ) = ∂T . More
generally, we can set Y = [ω]n and let I be the ideal of subsets of [ω]n which are not thick.
If m < n and T ⊆ [ω]n , we set ∂ (n−m) T = {s ∈ [ω]m : ∃t ∈ [ω]n−m (s ∪ t ∈ T )}. Then we can
set ψ(T ) = λ(n−m) ∂ (n−m) T .
Theorem 5.10. Let Y = [ω]n , let I be the thin ideal, and let ψ = λ(n−m) ∂ (n−m) for some
m < n. Suppose S ⊆ [ω]n is not (< ℵ0 )-near-closed, and say p, q ∈ β[ω]n where p intertwines
n
n
q. Then πm
(p) = πm
(q).
n
n
Proof. Suppose towards a contradiction that p0 := πm
(p) 6= q 0 := πm
(q) as witnessed by
m
0
m
0
A ⊆ [ω] with A ∈ p , [ω] \ A ∈ q . Then setting B := {s ∈ [ω]n : [s]m ⊆ A} and
C := {s ∈ [ω]n : [s]m ⊆ [ω]m \ A}, we have B ∈ p, C ∈ q, and B ∩ C = ∅. Note that both B
and C are ψ-closed. Since p and q are intertwined, we have ψ(B ∩ S) \ S ∈ q, so in particular
B \ S ∈ q. But since C ∈ q, this is a contradiction.


The next theorem along with Theorems 5.10 and 5.7 will prove Theorem 1.4.
Theorem 5.11. With I and ψ as in Theorem 5.10, there is S ⊆ [ω]n which is not (< ℵ0 )near-closed.
Proof. The following elegant proof is due to Anton Bernshteyn.
We take S to be the random n-uniform hypergraph. Suppose towards a contradiction that
S was k-near-closed for some k < ω. We write S = S0 ∪ · · · ∪ Sk−1
S with each Si near-closed.
Let Ti ⊆ [ω]n be a ψ-closed set with Si ∆Ti ∈ I, and write T = i<k Ti . So S∆T ∈ I. This
means that there is some ` < ω so that the hypergraph S∆T contains no clique of size `.
We now compute an upper bound on the number of induced subgraphs of S that can
appear on N vertices V := {v0 , ..., vN −1 } ⊆ ω. Since S is the random n-uniform hypergraph,
N
there must be 2( n ) many possibilities. But by assumption, S = T ∆G, where G is some
hypergraph with no cliques of size `. Since an induced subgraph of a ψ-closed graph is
N
ψ-closed, each Ti |V is determined by ∂ (n−m) (Ti |V ), so in particular, there are at most 2(m)
N
many possibilities for each Ti |V , so at most 2k(m) possibilities for T |V . As for G, we need
an estimate on the number of `-free n-uniform hypergraphs on N vertices.
It is a fact that

N
for some constant c > 0 depending only on ` and n, we can find c n subsets of N of size
` which pairwise have intersection smaller than n. By a probabilistic argument, it follows
that the proportion of n-uniform hypergraphs on N vertices which are `-free is at most
`

N

`

N

N

(1 − 2−(n) )c( n ) ≤ 2−c(n)( n ) := 2−d( n ) .
Multiplying together the number of choices for T |V with the number of choices for G|V , we
have that the number of possibilities for S|V is at most
N
N
N
(2(1−d)( n ) )(2k(m) )  2( n ) .

This shows that S is not (< ℵ0 )-near-closed.



Let us now briefly discuss why Theorem 1.4 implies Theorem 1.3. Recall (see [HS]) that
in any compact left-topological semigroup S, the smallest ideal K(S) is both the union of
the minimal right ideals and the union of the minimal left ideals. The intersection of any

18

D. BARTOŠOVÁ AND A. ZUCKER

minimal right ideal and any minimal left ideal is a group, so in particular contains exactly
one idempotent. More concretely, if M ⊆ S is a minimal right ideal and u ∈ M is an
idempotent, then Su is a minimal left ideal and M u = M ∩ Su. All the groups formed in
this way are algebraically isomorphic. When S = S(G) for some topological group G, we
can interpret this group as aut(M (G)), the group of G-flow isomorphisms of M (G).
Fix M ⊆ S(G) a minimal subflow, and let ϕ : S(G) → M be a G-map. Letting p = ϕ(1G ),
then we must have ϕ = λp . It follows that ϕ is a retraction iff ϕ = λu for some idempotent
u ∈ M . Furthermore, if p ∈ M , then there is a unique idempotent u ∈ M with p = pu ∈ M u.
It follows that for some q ∈ M we have λq ◦ λp = λu .
Now suppose N ⊆ S(G) is another minimal right ideal, and that x 6= y ∈ S(G) can be
separated by a retraction ψ onto N . Pick any p ∈ M and form the G-map λp ◦ ψ. Notice
that λp |N is an isomorphism. For some q ∈ M we have λp ◦ ψ = λq . Then for some r ∈ M ,
we have λr ◦ λq = λu a retraction. It follows that x and y are also separated by λu . Hence
the relation of being separated by a retraction does not depend on the choice of minimal
subflow M ⊆ S(G).
Now let G = S∞ , and let M 6= N be the minimal right ideals found in Theorem 1.4. Let
L be any minimal left ideal, and let u ∈ M ∩ L and v ∈ N ∩ L be idempotents. We will
show that u and v cannot be separated by retractions, so let ϕ : S(G) → M be a retraction.
Then ϕ = λw for some idempotent w ∈ M . Then ϕ(u) = wu = u since idempotents in M
are left identities for M . But now consider ϕ(v) = wv. By our assumption on M and N , wv
is an idempotent. However, we must also have wv ∈ M ∩ L since M and L are respectively
right and left ideals. It follows that wv = u, so ϕ(u) = ϕ(v) as desired.

6. Proximal and Distal
The technique of finding M, N ⊆ S(G) minimal subflows with J(M ) ∪ J(N ) a semigroup
allows for a quick solution to Ellis’s problem for some Polish groups G.
Recall from the introduction that a pair of points x, y in a G-flow X is called proximal if
there is p ∈ E(X) with xp = yp and X is proximal if every pair of points is proximal. Now
suppose that M (G) is proximal, i.e. that M (G) is proximal. Then every element of M is an
idempotent; to see why, notice that it suffices to show that M ∩ L is a singleton whenever
L is a minimal left ideal. Indeed, suppose u 6= p ∈ M ∩ L, with u idempotent. Suppose
that (u, p) were proximal, i.e. that for some q ∈ S(G) we have uq = pq. Since M ∩ L is a
group with identity u, we must have pu = p. Now as M is a minimal right ideal, find r ∈ M
with uqr = u. But then pqr = puqr = pu = p. This is a contradiction, so (u, p) cannot be
proximal.
A G-flow X is distal if every pair of non-equal points is distal, that is, not proximal. A
useful fact (see [A]) is that X is distal iff E(X) is a group. If M (G) is distal and M ⊆ S(G)
is a minimal subflow, then J(M ) is a singleton. To see this, note that if u, v ∈ J(M ), then
uv = vv = v, so (u, v) is a proximal pair. If u ∈ J(M ) is the unique idempotent, then the
map ϕ : E(M ) → M given by p → u · p is a G-flow isomorphism.
For automorphism groups G with M (G) proximal or distal, it follows that the conclusion
of Theorem 1.4 is automatic for any two minimal right ideals M 6= N . The same argument
for S∞ shows that any two idempotents of the same minimal left ideal cannot be separated
by retractions. Of course, we need to know that S(G) contains more than one minimal right
ideal; see ([Ba], Corollary 11) for a proof of this fact.

SAMUEL COMPACTIFICATIONS

19

The following theorem collects some examples of Polish groups G with M (G) proximal.
Theorem 6.1. Let G be either Homeo(2ω ) or the automorphism group of the countablyinfinite-dimensional vector space over a finite field. Then S(G) ∼
6= E(M (G))
We now consider the case when M (G) is distal. The rest of this section is spent proving
Theorem 1.5.
proof of Theorem 1.5. Suppose M (G) is distal and metrizable, and fix minimal M ⊆ S(G).
As M ∼
= E(M ) is a group, we see that Aut(M ), the group of G-flow isomorphisms of M , acts
transitively on M . Therefore by ch. 2, Theorem 13 of [A], M must be equicontinuous, hence
a compact metrizable group. Furthermore, as M must have a comeager orbit by [BYMT],
we have that M is a single orbit.
Let H = {g ∈ G : ug = u}. By [MNT], H is extremely amenable. To show that H is
normal, we show that the map ϕ : G → M given by ψ(g) = ug is a homomorphism. Indeed,
we have ug · uh = (ugu)h = ugh since u is a two-sided identity for M . This shows the
existence of the short exact sequence from item (2) of Theorem 1.5.
i

ϕ

Now suppose 1 → H →
− G−
→ K → 1 is a short exact sequence of groups with H extremely
amenable and K compact metrizable. Notice that the action k · g := k · ϕ(g) turns K into a
G-flow. Notice that ϕ is left uniformly continuous, so we can extend to a map ϕ : S(G) → K
via ϕ(p) = limgi →p ϕ(gi ). Let M ⊆ S(G) be minimal; then ϕ|M is a G-map.
Since H ⊆ G is extremely amenable, let u ∈ M be an H-fixed point. Now viewing K as
the right coset space {Hg : g ∈ G}, we build a G-equivariant map ψ : K → M by setting
ψ(Hg) = ug. As the image of ψ is dense in M , we will be done once we show that ψ is
continuous. To see this, notice that we have the following commutative diagram.
M
ψ

ϕ

K

λϕ(u)

K
Indeed, let Hg ∈ K. Then ϕ(ψ(Hg)) = ϕ(ug)) = ϕ(u) · g = ϕ(u) · Hg. Since λϕ(u) and ϕ
are continuous, we must have that ψ is continuous.


7. Some ultrafilters on [ω]2
This last section includes a short discussion of some ultrafilters motivated by the work in
section 5. The first main theorem of this section provides a counterpoint to Theorem 1.4.
Theorem 7.1. It is consistent with ZFC that there is a minimal subflow M ⊆ S(G) so that
if N ⊆ S(G) is a minimal subflow with J(M ) ∪ J(N ) a semigroup, then M = N .
The second theorem points out a key difference between selective ultrafilters and (2, 3)selective ultrafilters. Recall that if p, q ∈ βω, then we say that q ≥RK p if there is a function
f : ω → ω with f (q) = p. Another characterization of selective ultrafilters is that they are

20

D. BARTOŠOVÁ AND A. ZUCKER

exactly the ultrafilters which are minimal in the Rudin-Keisler order. The next theorem
shows that (2, 3)-selectives can be very far from Rudin-Keisler minimal.
Theorem 7.2. If p ∈ βω, there is a countably closed forcing extension P adding a (2, 3)selective ultrafilter q with q ≥RK p.
As it turns out, these two theorems will both be proven using the same forcing construction.
We define a forcing P which is very similar to a forcing defined by Laflamme [L]. A slightly
more straightforward forcing would suffice for Theorem 7.1 where we don’t refer to a fixed
p ∈ βω, but with a bit more work, we can prove both theorems.
F
Definition 7.3. Fix p ∈ βω. Write ω = n En with |En | = n. We define P = hP, ≤i as
follows.
(1) A condition A ∈ P is a subset of ω so that for every k < ω, we have {n < ω :
|A ∩ En | ≥ k} ∈ p.
(2) We declare that B ≤ A iff B ⊆ A.
If A, B ∈ P, we define B  A iff there is k < ω so that {m < ω : |Em ∩ (B \ A)| ≤ k} ∈ p.
It is straightforward to see that hP, i is a separative pre-order which is equivalent to P.
Lemma 7.4. P is countably closed.
Proof. T
First notice that if hAn : n < ωi is a -decreasing sequence in P , then setting
A0n = i≤n Ai , we have that A0n is -equivalent to An . So we may freely work with ≤decreasing sequences.
Suppose hAn : n < ωi is a ≤-decreasing sequence in P . Write S(m, k) = {n < ω :
|Am ∩ En | ≥ k}. Note that S(m, k) ∈ p for every m, k < ω. Also, if m ≤ m0 and k ≤ k 0 ,
then S(m0 , k 0 ) ⊆ S(m, k).
S For m ≥ 1, we define Tm = S(m, m) \ S(m + 1, m + 1). Note that if m < ω, then
n≥m Tm = S(m, m). If m ≥ 1 and n ∈ Tm , then |Am ∩ En | ≥ m. We form B ∈ P by setting
[ [
B=
Am ∩ En .
m≥1 n∈Tm

For each m ≥ 1, we have {n < ω : |B ∩ En | ≥ m} = S(m, m) ∈ p, so B ∈ p. To see that
B  Am , we note that {n < ω : B 6⊆ Am } ⊆ ω \ S(m, m).

If A ∈ P, we set Ã =

S

n [A

∩ En ]2 ⊆ [ω]2 . The next proposition will prove Theorem 7.2.

Proposition 7.5. Let G ⊆ P be generic. Then G̃ := {Ã : A ∈ G} generates a thick ultrafilter
on [ω]2 which is (2, n)-selective for every n. Furthermore, this ultrafilter is RK-above p.
Proof. Set E2 = 1eP , and suppose E2 = S t T . Let A ∈ p. By Ramsey’s theorem, there is
some non-decreasing function k → b(2, k) increasing to infinity so that any 2-coloring of the
complete graph on k vertices has a monochromatic clique of size b(2, k). If |A ∩ EN | = k,
then let XN ⊆ A ∩ EN be chosen so that |XN | = b(2, k) and X˜N ⊆ S or X˜N ⊆ T . Define
S 0 , T 0 ⊆ ω, placing N ∈ S 0 S
or N ∈ T 0 depending on which outcome happens. WLOG suppose
0
S ∈ p. Then letting X = N ∈S 0 XN , we have X ∈ P, X ≤ A, and X decides whether S or
T is in the filter generated by G̃.

SAMUEL COMPACTIFICATIONS

21

The argument that the ultrafilter generated by G̃ is (2, n)-selective is almost the exact
same. By Ramsey’s theorem, there is some non-decreasing function k → b(n, k) increasing
to infinity so that any 2-coloring of the complete n-uniform hypergraph on k-vertices has a
monochromatic clique of size b(n, k). Now letting En = λ(n−2) (E2 ), fix a partition En = StT .
If A ∈ P, we can in a similar fashion find X ≤ A deciding whether S or T is in the filter
λ(n−2) (G̃).
Lastly, let ψ : E2 → ω be so that ψ({x, y}) = n iff {x, y} ⊆ En . Then if U ∈ V [G] is the
ultrafilter generated by G̃, then ψ(U) = p.

We now turn towards the proof of Theorem 7.1. To do this, we use Theorem 5.7. Working
in V [G], let MG ⊆ S(S∞ ) be the unique minimal subflow so that ϕ(F2MG ) is the ultrafilter
generated by G̃. We need to show that {A(<0 , <1 ) : (<0 , <1 ) ∈ AMG } generates G̃. To see
why this is, fix A ∈ P. We may assume that if A ∩ En 6= ∅, then |A ∩ En | ≥ 2. We will
construct linear orders
S <0 and <1 so that A(<0 , <1 ) = Ã.
First write ω = n Xn , where X0 = ω \ A and Xn = A ∩ En . Some of the Xn may
be empty, but this is fine. First define <0 and <1 on X0 to be any linear orders which
completely disagree. Suppose <0 and <1 have been defined on X0 ∪ · · · ∪ Xn−1 . First define
<0 and <1 on Xn so that they agree. Now place Xn <0 -below everything built so far and
also <1 -above everything built so far. Then A(<0 , <1 ) = Ã as desired. This completes the
proof of Theorem 7.1.
The proof of Theorem 7.1 suggests another type of ultrafilter on [ω]2 we can define. If
p ∈ β[ω]2 is thick, define Ap = {(<0 , <1 ) : A(<0 , <1 ) ∈ p}. As we saw in section 5, Ap is an
equivalence relation on LO(ω).
Definition 7.6. Let p ∈ β[ω]2 be a thick ultrafilter. We call p a linear order ultrafilter if
{A(<0 , <1 ) : (<0 , <1 ) ∈ Ap } generates p. Call p a weak linear order ultrafilter if p is the
unique thick ultrafilter containing every A(<0 , <1 ) with (<0 , <1 ) ∈ p.
One can prove that there are thick ultrafilters p ∈ β[ω]2 which are not weak linear order
ultrafilters, providing an alternate proof of Theorem 1.4. The proof is very similar to the
proof that some p ∈ β[ω]2 is not weakly (2, 3)-selective.
We end with some open question about these ultrafilters.
Question 7.7. Does ZFC prove the existence of (2, 3)-selective ultrafilters? Of linear order
ultrafilters?
Question 7.8. Can there exist a weakly (2, 3)-selective ultrafilter which is not (2, 3)-selective?
Same question for linear order ultrafilters.
The last question is motivated by Theorem 7.2. This shows that (2, 3)-selective ultrafilters
can exist arbitrarily high up in the Rudin-Keisler order.
Question 7.9. Is it consistent with ZFC that the (2, 3)-selective ultrafilters are upwards
Rudin-Keisler cofinal?
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