aa r X i v : . [ m a t h . L O ] F e b LACUNARY SETS FOR ACTIONS OF TSI GROUPS
BENJAMIN D. MILLER
Abstract.
Under a mild definability assumption, we characterizethe family of Borel actions Γ y X of tsi Polish groups on Po-lish spaces that can be decomposed into countably-many actionsadmitting complete Borel sets that are lacunary with respect toan open neighborhood of 1 Γ . In the special case that Γ is non-archimedean, it follows that there is such a decomposition if andonly if there is no continuous embedding of E N into E X Γ . Introduction
The orbit equivalence relation induced by a group action Γ y X isthe equivalence relation on X given by x E X Γ y ⇐⇒ ∃ γ ∈ Γ γ · x = y .More generally, the orbit relation associated with a set ∆ ⊆ Γ is thebinary relation on X given by x R X ∆ y ⇐⇒ ∃ δ ∈ ∆ δ · x = y . A set Y ⊆ X is ∆ -lacunary if y R X ∆ z = ⇒ y = z for all y, z ∈ Y .Following the usual abuse of language, we say that an equivalencerelation E on X is countable if | [ x ] E | ≤ ℵ for all x ∈ X . We say that aset Y ⊆ X is E -complete if [ x ] E ∩ Y = ∅ for all x ∈ X . The product ofequivalence relations E n on X n is the equivalence relation Q n ∈ N E n on Q n ∈ N X n given by ( x n ) n ∈ N ( Q n ∈ N E n ) ( y n ) n ∈ N ⇐⇒ ∀ n ∈ N x n E n y n .The N -fold power of E is given by E N = Q n ∈ N E .A graph on X is an irreflexive symmetric set G ⊆ X × X . We saythat a set Y ⊆ X is G -independent if G ↾ Y = ∅ . A Z -coloring of G isa map π : X → Z such that π − ( { z } ) is G -independent for all z ∈ Z .A homomorphism from a binary relation R on X to a binary relation S on Y is a map φ : X → Y such that w R x = ⇒ φ ( w ) S φ ( x ) for all w, x ∈ X . More generally, a homomorphism from a sequence ( R i ) i ∈ I ofbinary relations on X to a sequence ( S i ) i ∈ I of binary relations on Y isa map φ : X → Y that is a homomorphism from R i to S i for all i ∈ I .A reduction of R to S is a homomorphism from ( R, ∼ R ) to ( S, ∼ S ),and an embedding of R into S is an injective reduction of R to S . Mathematics Subject Classification.
Primary 03E15, 28A05.
Key words and phrases.
Essentially countable, lacunary set, reducibility.The author was supported in part by FWF Grants P28153 and P29999.
Suppose that Γ is a Polish group and X is a Borel space. We say thata Borel action Γ y X is σ -lacunary if there are E X Γ -invariant Borel sets X n ⊆ X with the property that X = S n ∈ N X n , open neighborhoods∆ n ⊆ Γ of 1 Γ , and ∆ n -lacunary E X n Γ -complete Borel sets B n ⊆ X n forall n ∈ N . A Borel equivalence relation on a standard Borel space is essentially countable if it is Borel reducible to a countable Borel equiv-alence relation on a standard Borel space. The Lusin-Novikov uni-formization theorem (see, for example, [Kec95, Theorem 18.10]) easilyimplies that if X is a standard Borel space, Γ y X is a σ -lacunaryBorel action, and E X Γ is Borel, then E X Γ is essentially countable.A well-known example of a non-essentially-countable Borel equiva-lence relation is the N -fold power of the equivalence relation E on 2 N given by c E d ⇐⇒ ∃ n ∈ N ∀ m ≥ n c ( m ) = d ( m ).A topological group is non-archimedean if there is a neighborhoodbasis of the identity consisting of open subgroups. A topological groupis tsi if it has a compatible two-sided-invariant metric. Klee has shownthat a Hausdorff group is tsi if and only if there is a neighborhood basisof the identity consisting of conjugation-invariant open subsets (see[Kle52, 1.5]). It follows that a Hausdorff group is both non-archimedeanand tsi if and only if there is a neighborhood basis of the identityconsisting of normal open subgroups (see, for example, [GX14, § X is a Polish space, Γ y X is Borel, and E X Γ is Borel, theneither E X Γ is essentially countable or there is a continuous embeddingof E N into E X Γ (see [HK01, Theorem 8.1]). Our goal here is to give aclassical proof of the strengthening in which essential countability isreplaced with σ -lacunarity.Given a graph G on a Borel space X , we write χ B ( G ) ≤ ℵ to indicatethat G has countable Borel chromatic number , meaning that there isa Borel N -coloring of G . Kechris-Solecki-Todorcevic have shown thatthere is a minimal analytic graph G on a standard Borel space thatdoes not have countable Borel chromatic number (see [KST99, § §
1, we characterize the class of increasing-in- j sequences ( G i,j ) i,j ∈ N of analytic graphs for which there exist a function f : N → N and acontinuous homomorphism φ : 2 N → X from a sequence of pairwisedisjoint copies of G to ( G i,f ( i ) ) i ∈ N . In §
2, we show that for appro-priately chosen graphs, the inexistence of such homomorphisms yields σ -lacunarity. In §
3, we describe various ways of refining such homo-morphisms. And in §
4, we establish a characterization of σ -lacunarityfor Borel actions Γ y X of tsi Polish groups with the property that SSENTIAL COUNTABILITY 3 R X ∆ is Borel for every open set ∆ ⊆ Γ. In the special case that Γ isnon-archimedean, this yields our main result.1.
A graph-theoretic dichotomy
Fix k n ∈ N such that k = 0, ∀ n ∈ N k n +1 ≤ max { k m | m ≤ n } + 1,and ∀ k ∈ N ∃ ∞ n ∈ N k n = k , as well as s n ∈ n with the property that ∀ k ∈ N ∀ s ∈ < N ∃ n ∈ N ( k = k n and s ⊑ s n ).For all s ∈ < N , we use G s to denote the graph on 2 N given by G s = { ( s a ( i ) a c ) i< | c ∈ N } . For all k ∈ N , we use G ,k to denotethe graph on 2 N given by G ,k = S { G s n | k = k n and n ∈ N } . Theorem 1.1.
Suppose that X is a Hausdorff space and ( G i,j ) i,j ∈ N isan increasing-in- j sequence of analytic graphs on X . Then exactly oneof the following holds: (1) There are Borel sets B n ⊆ X such that X = S n ∈ N B n and ∀ n ∈ N ∃ i ∈ N ∀ j ∈ N χ B ( G i,j ↾ B n ) ≤ ℵ . (2) There exist a function f : N → N and a continuous homomor-phism φ : 2 N → X from ( G ,k ) k ∈ N to ( G k,f ( k ) ) k ∈ N .Proof. To see that conditions (1) and (2) are mutually exclusive, sup-pose that both hold, fix n ∈ N for which φ − ( B n ) is non-meager, fix i ∈ N such that ∀ j ∈ N χ B ( G i,j ↾ B n ) ≤ ℵ , fix a Borel coloring ψ : B n → N of G i,f ( i ) ↾ B n , fix m ∈ N for which ( φ − ◦ ψ − )( { m } )is non-meager, fix s ∈ < N for which ( φ − ◦ ψ − )( { m } ) is comeagerin N s , and fix ℓ ∈ N for which i = k ℓ and s ⊑ s ℓ . It only re-mains to observe that there are comeagerly many c ∈ N such that s ℓ a ( i ) a c ∈ ( φ − ◦ ψ − )( { m } ) for all i <
2, contradicting the factthat φ is a homomorphism from G s ℓ to G i,f ( i ) .It remains to show that at least one of conditions (1) and (2) holds.We can assume that G i,j = ∅ for all i, j ∈ N , in which case there arecontinuous surjections φ i,j : N N → G i,j for all i, j ∈ N , as well as acontinuous surjection φ X : N N → S i,j ∈ N proj X ( G i,j ).We will recursively define decreasing sequences ( X αi,j ) α<ω of subsetsof X such that X αi,j ⊆ X αi,j +1 and χ B ( G i,j ↾ ∼ X αi,j ) ≤ ℵ for all α < ω and i, j ∈ N . We begin by setting X i,j = X for all i, j ∈ N , and defining X λi,j = T α<λ X αi,j for all i, j ∈ N and limit ordinals λ < ω . To describethe construction of X α +1 i,j from X αi,j , we require several preliminaries.We say that a quadruple a = ( n a , f a , φ a , ( ψ an ) n Suppose that ( X i,j ) i,j ∈ N is a sequence of subsets of X and a is an approximation for which k n a ∈ dom( f a ) and A ( a, ( X i,j ) i,j ∈ N ) isnot G k na ,f a ( k na ) -independent. Then a is not ( X i,j ) i,j ∈ N -terminal.Proof. Fix configurations γ and γ , compatible with a and ( X i,j ) i,j ∈ N ,for which (( φ X ◦ φ γ i )( s n a )) i< ∈ G k na ,f a ( k na ) . Then there exists b ∈ N N such that φ k na ,f a ( k na ) ( b ) = (( φ X ◦ φ γ i )( s n a )) i< . Let γ be the config-uration given by n γ = n a + 1, f γ = f a , φ γ ( s a ( i )) = φ γ i ( s ) forall i < s ∈ n a , ψ γn ( s a ( i )) = ψ γ i n ( s ) for all i < n < n a ,and s ∈ n a − n − , and ψ γn a ( ∅ ) = b . Then the unique approximation b with which γ is compatible is a one-step extension of a , so a is not( X i,j ) i,j ∈ N -terminal. Lemma 1.3. Suppose that ( X i,j ) i,j ∈ N is a sequence of subsets of X , a is an approximation for which k n a / ∈ dom( f a ) , and there exists ℓ ∈ N such that A ( a, ( X i,j ) i,j ∈ N ) is not G k na ,ℓ -independent. Then a is not ( X i,j ) i,j ∈ N -terminal.Proof. Fix configurations γ and γ , compatible with a and ( X i,j ) i,j ∈ N ,for which (( φ X ◦ φ γ i )( s n a )) i< ∈ G k na ,ℓ . By increasing ℓ if necessary,we can assume that φ γ (2 n a ) ∪ φ γ (2 n a ) ⊆ X k na ,ℓ . Fix b ∈ N N suchthat φ k na ,ℓ ( b ) = (( φ X ◦ φ γ i )( s n a )) i< , and let γ be the configurationgiven by n γ = n a + 1, f γ ( k ) = f a ( k ) for all k < k n a , f γ ( k n a ) = ℓ , SSENTIAL COUNTABILITY 5 φ γ ( s a ( i )) = φ γ i ( s ) for all i < s ∈ n a , ψ γn ( s a ( i )) = ψ γ i n ( s ) forall i < n < n a , and s ∈ n a − n − , and ψ γn a ( ∅ ) = b . Then the uniqueapproximation b with which γ is compatible is a one-step extension of a , so a is not ( X i,j ) i,j ∈ N -terminal.As Lusin’s separation theorem (see, for example, [Kec95, Theorem14.7]) easily implies that every G i,j -independent analytic set is con-tained in a G i,j -independent Borel set, Lemmas 1.2 and 1.3 ensurethat if ( X i,j ) i,j ∈ N is a sequence of analytic sets and a is an ( X i.j ) i,j ∈ N -terminal approximation, then there is a Borel set B ( a, ( X i,j ) i,j ∈ N ) ⊇ A ( a, ( X i,j ) i,j ∈ N ) that is G k na ,f ( k na ) -independent if k n a ∈ dom( f a ), and G k na ,ℓ -independent for all ℓ ∈ N if k n a / ∈ dom( f a ).We finally define X α +1 k,ℓ to be the difference of X αk,ℓ and the union ofthe sets of the form B ( a, ( X αi,j ) i,j ∈ N ), where a is an ( X αi,j ) i,j ∈ N -terminalapproximation, k n a = k , and f a ( k n a ) ≥ ℓ if k n a ∈ dom( f a ). Lemma 1.4. Suppose that α < ω and a is an approximation that isnot ( X α +1 i,j ) i,j ∈ N -terminal. Then there is a one-step extension of a thatis not ( X αi,j ) i,j ∈ N -terminal.Proof. Fix a one-step extension b of a for which there is a configuration γ compatible with b and ( X α +1 i,j ) i,j ∈ N . Note that if k n b ∈ dom( f b ), then( φ X ◦ φ γ )( s n b ) ∈ X α +1 k nb ,f b ( k nb ) , so A ( b, ( X αi,j ) i,j ∈ N ) ∩ X α +1 k nb ,f b ( k nb ) = ∅ , thus b is not ( X αi,j ) i,j ∈ N -terminal. And if k n b / ∈ dom( f b ), then there exists ℓ ∈ N for which ( φ X ◦ φ γ )( s n b ) ∈ X α +1 k nb ,ℓ , so A ( b, ( X αi,j ) i,j ∈ N ) ∩ X α +1 k nb ,ℓ = ∅ ,thus b is not ( X αi,j ) i,j ∈ N -terminal.Fix α < ω such that the families of ( X αi,j ) i,j ∈ N -terminal approxi-mations and ( X α +1 i,j ) i,j ∈ N -terminal approximations are the same, let a denote the unique approximation a with the property that n a = 0,and observe that A ( a , ( X i,j ) i,j ∈ N ) = T i ∈ N S j ∈ N X i,j for all sequences( X i,j ) i,j ∈ N of subsets of X . In particular, it follows that if a is ( X αi,j ) i,j ∈ N -terminal, then T i ∈ N S j ∈ N X α +1 i,j = ∅ , so condition (1) holds.Otherwise, by recursively applying Lemma 1.4, we obtain one-stepextensions a n +1 of a n that are not ( X αi,j ) i,j ∈ N -terminal for all n ∈ N . Define f : N → N by f = S n ∈ N f a n , define φ : 2 N → N N by φ ( c ) = S n ∈ N φ a n ( c ↾ n ) for all c ∈ N , and define ψ n : 2 N → N N by ψ n ( c ) = S m ∈ N ψ a n +1+ m n ( c ↾ m ) for all c ∈ N and n ∈ N . To see that φ X ◦ φ is a homomorphism from ( G ,k ) k ∈ N to ( G k,f ( k ) ) k ∈ N , we will showthat ( φ k n ,f ( k n ) ◦ ψ n )( c ) = (( φ X ◦ φ )( s n a ( i ) a c )) i< for all c ∈ N and n ∈ N . For this, it is sufficient to show that if U ⊆ X × X is anopen neighborhood of ( φ k n ,f ( k n ) ◦ ψ n )( c ) and V ⊆ X × X is an open B.D. MILLER neighborhood of (( φ X ◦ φ )( s n a ( i ) a c )) i< , then U ∩ V = ∅ . To-wards this end, fix m ∈ N for which φ k n ,f ( k n ) ( N ψ an +1+ mn ( s ) ) ⊆ U and Q i< φ X ( N φ an +1+ m ( s n a ( i ) a s ) ) ⊆ V , where s = c ↾ m . The fact that a m isnot ( X αi,j ) i,j ∈ N -terminal then yields a configuration γ compatible with a m , so ( φ k n ,f ( k n ) ◦ ψ γn )( s ) ∈ U and (( φ X ◦ φ γ )( s n a ( i ) a s )) i< ∈ V ,thus U ∩ V = ∅ . 2. Lacunary sets Here we note the connection between condition (1) of Theorem 1.1and lacunary sets. Proposition 2.1. Suppose that Γ is a tsi analytic Hausdorff group, X is an analytic Hausdorff space, Γ y X is a σ -lacunary Borel actionsuch that R X ∆ is Borel for all open sets ∆ ⊆ Γ , (∆ i ) i ∈ N is a neighborhoodbasis of Γ consisting of conjugation-invariant symmetric open sets, and G i,j = R X ∆ i \ R X ∆ j for all i, j ∈ N . Then there are Borel sets B n ⊆ X such that X = S n ∈ N B n and ∀ n ∈ N ∃ i ∈ N ∀ j ∈ N χ B ( G i,j ↾ B n ) ≤ ℵ .Proof. By breaking X into countably-many E X Γ -invariant Borel sets,we can assume that there is an open neighborhood ∆ ⊆ Γ of 1 Γ forwhich there is a ∆-lacunary E X Γ -complete Borel set B ⊆ X .Fix i ∈ N for which there is an open neighborhood ∆ ′ ⊆ Γ of 1 Γ such that (∆ ′ ) − ∆ i ∆ ′ ⊆ ∆. To see that χ B ( G i,j ) ≤ ℵ for all j ∈ N ,fix j ∈ N and an open set ∆ ′′ ⊆ ∆ ′ such that ∆ ′′ (∆ ′′ ) − ⊆ ∆ j . Lemma 2.2. The set ∆ ′′ B is G i,j -independent.Proof. Suppose that x ′′ , y ′′ ∈ ∆ ′′ B are R X ∆ i -related. Then there exist δ ′′ x , δ ′′ y ∈ ∆ ′′ for which the points x = ( δ ′′ x ) − · x ′′ and y = ( δ ′′ y ) − · y ′′ arein B . As x and y are R X (∆ ′′ ) − ∆ i ∆ ′′ -related, so R X ∆ -related, thus equal,it follows that x ′′ and y ′′ are R X ∆ ′′ (∆ ′′ ) − -related, thus R X ∆ j -related.The conjugation invariance of ∆ i and ∆ j now ensures that γ ∆ ′′ B is G i,j -independent, and therefore contained in an G i,j -independent Borelset, for all γ ∈ Γ. As X is the union of countably-many sets of thisform, it follows that χ B ( G i,j ) ≤ ℵ .A topological group is cli if it has a compatible complete left-invariantmetric, or equivalently, a compatible complete right-invariant metric(see, for example, [Bec98, Proposition 3.A.2]). It is well-known thatevery tsi group is cli (see, for example, [BK96, Corollary 1.2.2]). Proposition 2.3. Suppose that Γ is a cli Polish group, X is an analyticmetric space, Γ y X is continuous, (∆ i ) i ∈ N is a neighborhood basis of SSENTIAL COUNTABILITY 7 Γ consisting of symmetric open sets, G i,j = R X ∆ i \ R X ∆ j for all i, j ∈ N ,and there are Borel sets B n ⊆ X with the property that X = S n ∈ N B n and ∀ n ∈ N ∃ i ∈ N ∀ j ∈ N χ B ( G i,j ↾ B n ) ≤ ℵ . Then Γ y X is σ -lacunary.Proof. We can assume that Γ is not discrete, since otherwise Γ y X is trivially σ -lacunary. So by passing to a subsequence of (∆ i ) i ∈ N , wecan also assume that ∆ i +12 ⊆ ∆ i for all i ∈ N . By breaking each B n into countably-many Borel sets, we can moreover assume that thereare natural numbers i n ∈ N such that B n is G i n ,i n +3 -independent and χ B ( G i n ,i n +4+ j ↾ B n ) ≤ ℵ for all j, n ∈ N . As a result of Montgomer-y-Novikov ensures that the class of Borel sets is closed under categoryquantification (see, for example, [Kec95, Theorem 16.1]), it follows thatthe map φ : X → N given by φ ( x ) = min { n ∈ N | ∃ ∗ γ ∈ Γ γ · x ∈ B n } is Borel. By passing to the E X Γ -invariant Borel sets X n = φ − ( B n ), it issufficient to show that if i ∈ N and there is a G i,i +3 -independent Borelset B ⊆ X with the property that ∀ j ∈ N χ B ( G i,i +4+ j ↾ B ) ≤ ℵ and ∀ x ∈ X ∃ ∗ γ ∈ Γ γ · x ∈ B , then there is a ∆ i +2 -lacunary E X Γ -completeBorel set.Towards this end, observe that the set E = R X ∆ i +3 ↾ B is an equiv-alence relation. As E has countable index below E X Γ ↾ B , by thinningdown B if necessary, we can assume that ∀ x ∈ B ∃ ∗ γ ∈ Γ x E γ · x . Fixpositive real numbers ǫ j → 0, as well as Borel colorings c i +4+ j : B → N of G i,i +4+ j ↾ B such that diam c − i +4+ j ( { m } ) ≤ ǫ j for all j, m ∈ N . Foreach j ∈ N and x ∈ B , let s i +4+ j ( x ) denote the lexicographically min-imal sequence s ∈ N j +1 for which there are non-meagerly many γ ∈ Γwith the property that γ · x ∈ T k ≤ j c − i +4+ k ( { s ( k ) } ) ∩ [ x ] E , and let C i +4+ j denote the set of x ∈ B for which s i +4+ j ( x ) = ( c i +4+ k ( x )) k ≤ j .A ray from x ∈ B through ( C i +4+ j ) j ∈ N is a sequence ( δ i +3+ j ) j ∈ N withthe property that δ i +3+ j ∈ ∆ i +3+ j and δ i +3+ j · · · δ i +3 · x ∈ C i +4+ j for all j ∈ N . A straightforward recursive construction yields the existence ofsuch rays, while a straightforward inductive argument ensures that if( δ i +3+ j ) j ∈ N is such a ray, then δ i +3+ k · · · δ i +3+ j ∈ ∆ i +2+ j for all k > j .In particular, it follows that ( δ i +3+ j · · · δ i +3 ) j ∈ N is Cauchy with respectto every compatible complete right-invariant metric on Γ, and thereforeconverges to some δ ∈ ∆ i +2 .Observe now that if ( δ xi +3+ j ) j ∈ N and ( δ yi +3+ j ) j ∈ N are rays from points x and y in B through ( C i +4+ j ) j ∈ N , and δ x and δ y are the correspondinglimit points, then δ x · x R X ∆ i +2 δ y · y = ⇒ x R X ∆ i y = ⇒ x E y and x E y = ⇒ δ x · x = δ y · y . We therefore obtain a function ψ : B → X by insisting that ψ ( x ) = y if and only if there is a ray ( δ i +3+ j ) j ∈ N from x through ( C i +4+ j ) j ∈ N for which δ i +3+ j · · · δ i +3 · x → y . It also follows B.D. MILLER that the corresponding set ψ ( B ) is ∆ i +2 -lacunary, and the fact that ∀ y ∈ ψ ( B ) ∃ ∗ γ ∈ Γ ψ ( γ · y ) = y ensures that ψ ( B ) is Borel.3. Compositions Here we note several ways of refining condition (2) of Theorem 1.1. Proposition 3.1. Suppose that f : N → N . Then there is a continuoushomomorphism φ : 2 N → N from ( G ,k ) k ∈ N to ( G ,f ( k ) ) k ∈ N .Proof. Recursively construct m n ∈ N and u n ∈ < N with the propertythat k m n = f ( k n ) and s m n = φ n ( s n ), where φ n : 2 n → m n is given by φ n ( t ) = u a L i 2. As k m n = f ( k n ), itfollows that φ ( s n a (0) a c ) G ,f ( k n ) φ ( s n a (1) a c ).For all s, t ∈ < N , we use G s,t to denote the subgraph of G s given by G s,t = { ( s a ( i ) a t a c ) i< | c ∈ N } . Proposition 3.2. Suppose that ( R i,j ) i,j ∈ N is a sequence of analyticbinary relations on N with the property that G ,k ⊆ S j ∈ N R i,j for all i, k ∈ N . Then there are functions g n : 2 2, thus ( φ [0 , ∞ ) ( s n a ( i ) a t a c )) i< ∈ R k n +1+ | t | ,g n +1+ | t | ( t ) .For all F ⊆ N × N and c, d ∈ F , let ∆( c, d ) be the set of ( m, n ) ∈ F with c ( m, n ) = d ( m, n ). For all i ∈ N , set ∆ i ( c, d ) = ∆( c, d ) ∩ ( i × N ).When F ∈ [ i × N ] < ℵ , set D i,F = { ( c, d ) ∈ N × N × N × N | ∆ i ( c, d ) = F } . Proposition 3.3. Suppose that i ∈ N , F ∈ [ i × N ] < ℵ , R ⊆ D i,F hasthe Baire property, and there are densely many u ∈ < ( N × N ) for whichthere is a homeomorphism φ : N u → N u whose graph is contained in D i, ∅ \ RR − . Then R is meager.Proof. Suppose, towards a contradiction, that R is non-meager. Thenthere exist G ∈ [( i × N ) \ F ] < ℵ and H, H ′ ∈ [( N \ i ) × N ] < ℵ for whichthere exist r ∈ F , s ∈ G , t ∈ H , and t ′ ∈ H ′ with the propertythat R is comeager in D i,F ∩ ( N r ∪ s ∪ t × N r ∪ s ∪ t ′ ), in which case the set S of ( c, ( d, d ′ )) ∈ ( i × N ) \ ( F ∪ G ) × (2 (( N \ i ) × N ) \ H × (( N \ i ) × N ) \ H ′ ) with theproperty that (( c ∪ r ∪ s ) ∪ ( d ∪ t )) R (( c ∪ r ∪ s ) ∪ ( d ′ ∪ t ′ )) is comeager.Let C denote the set of c ∈ ( i × N ) \ ( F ∪ G ) for which S c is comeager,and let D denote the set of ( c, d ) ∈ ( i × N ) \ ( F ∪ G ) × (( N \ i ) × N ) \ H for which( S c ) d is comeager. The Kuratowski-Ulam theorem ensures that C iscomeager, as is D c for all c ∈ C , thus D c × D c ⊆ S c S − c for all c ∈ C .Fix I ∈ [( i × N ) \ ( F ∪ G )] < ℵ and J ∈ [(( N \ i ) × N ) \ H ] < ℵ forwhich there exist u ∈ I and v ∈ J with the property that thereis a homeomorphism φ : N ( r ∪ s ∪ u ) ∪ ( t ∪ v ) → N ( r ∪ s ∪ u ) ∪ ( t ∪ v ) whose graph iscontained in D i, ∅ \ RR − . Fix c ∈ C ∩ N u and define ψ : 2 (( N \ i ) × N ) \ H ∩N v → (( N \ i ) × N ) \ H ∩N v by ψ ( d ) = (proj (( N \ i ) × N ) \ H ◦ φ )(( c ∪ r ∪ s ) ∪ ( d ∪ t ))for all d ∈ (( N \ i ) × N ) \ H ∩ N v . The fact that ψ is a homeomorphism thenensures that there are comeagerly many d ∈ (( N \ i ) × N ) \ H ∩ N v that arealso in D c ∩ ψ − ( D c ). But the defining property of φ ensures that d and ψ ( d ) are not ( S c S − c )-related, the desired contradiction. For each i ∈ N , define δ i : 2 N × N × N × N → N ∪ {ℵ } by setting δ i ( c, d ) = | ∆( c, d ) ∩ ( { i } × N ) | for all c, d ∈ N × N .A homomorphism from a function f : X × X → N to a function g : Y × Y → N is a map φ : X → Y such that f ( w, x ) = g ( φ ( w ) , φ ( x ))for all w, x ∈ X . More generally, a homomorphism from a sequence( f i : X × X → N ) i ∈ I to a sequence ( g i : Y × Y → N ) i ∈ I is a map φ : X → Y that is a homomorphism from f i to g i for all i ∈ I . Proposition 3.4. Suppose that C ⊆ N × N is comeager. Then there isa continuous homomorphism φ : 2 N × N → C from ( δ i ) i ∈ N to ( δ i ) i ∈ N .Proof. Fix dense open sets U n ⊆ N × N for which T n ∈ N U n ⊆ C . Lemma 3.5. For all F, G ∈ [ N × N ] < ℵ , φ : 2 F → G , and n ∈ N , thereexist H ∈ [ ∼ G ] < ℵ and t ∈ H such that N φ ( s ) ∪ t ⊆ U n for all s ∈ F .Proof. Fix an enumeration ( s m ) m< | F | of 2 F , and recursively find pair-wise disjoint sets H m ∈ [ ∼ G ] < ℵ and t m ∈ H m with N φ ( s m ) ∪ S ℓ ≤ m t ℓ ⊆ U n for all m < | F | . Define H = S m< | F | H m and t = S m< | F | t m .Fix an injective enumeration ( i n , j n ) n ∈ N of N × N , and for all n ∈ N ,set F n = { ( i m , j m ) | m < n } . By recursively appealing to Lemma3.5, we obtain H n ∈ [ N × N ] < ℵ and j ′ n ∈ ∼ ( H n ) i n for which the sets G n = H n ∪ { ( i n , j ′ n ) } are pairwise disjoint, as well as t n ∈ H n such that N φ n ( s ) ∪ t n ⊆ U n for all n ∈ N and s ∈ F , where φ n : 2 F n → S m Suppose that D ⊆ N × N × N × N is closed and nowheredense in D i,F for all i ∈ N and F ∈ [ i × N ] < ℵ , and R ⊆ N × N × N × N is meager in D i,F for all i ∈ N and F ∈ [ i × N ] < ℵ . Then there is acontinuous homomorphism φ : 2 N × N → N × N from (∆(2 N ) k × E N ) k ∈ N to (∆(2 N ) k × E N ) k ∈ N that is also a homomorphism from ( ∼ ∆(2 N × N ) , ∼ E N ) to ( ∼ D, ∼ R ) . SSENTIAL COUNTABILITY 11 Proof. For all i ∈ N and F ∈ [ i × N ] < ℵ , fix a decreasing sequence( U i,F,n ) n ∈ N of dense open symmetric subsets of D i,F \ D whose intersec-tion is disjoint from R . Lemma 3.7. For all F, G ∈ [ N × N ] < ℵ , φ : 2 F → G , and i, n ∈ N , there exist H ∈ [ ∼ G ] < ℵ and t , t ∈ H with the property that ∆ i ( t , t ) = ∅ and D i, ∆ i ( φ ( s ) ,φ ( s )) ∩ Q k< N φ ( s k ) ∪ t k ⊆ U i, ∆ i ( φ ( s ) ,φ ( s )) ,n for all s , s ∈ F .Proof. Fix an enumeration ( s ,m , s ,m ) m< | F | of 2 F × F , and recursivelyfind pairwise disjoint sets H m ∈ [ ∼ G ] < ℵ and t ,m , t ,m ∈ H m such that∆ i ( t ,m , t ,m ) = ∅ and D i, ∆ i ( φ ( s ,m ) ,φ ( s ,m )) ∩ Q k< N φ ( s k,m ) ∪ S ℓ ≤ m t k,ℓ ⊆ U i, ∆ i ( φ ( s ,m ) ,φ ( s ,m )) ,n for all m < | F | . Set H = S m< | F | H m and t k = S m< | F | t k,m .Fix an injective enumeration ( i n , j n ) n ∈ N of N × N , and for all n ∈ N ,set F n = { ( i m , j m ) | m < n } . By recursively appealing to Lemma 3.7,we obtain pairwise disjoint sets G n ∈ [ N × N ] < ℵ and t ,n , t ,n ∈ G n such that ∆ i n ( t ,n , t ,n ) = ∅ and D i n , ∆ in ( φ n ( s ) ,φ n ( s )) ∩ Q k< N φ n ( s k ) ∪ t k,n ⊆ U i n , ∆ in ( φ n ( s ) ,φ n ( s )) ,n for all n ∈ N and s , s ∈ F n , where φ n : 2 F n → S m We will abuse notation by identifying ∆(2 N ) k × E × ∆(2 N ) N withthe corresponding equivalence relation on 2 N × N for all k ∈ N . Theorem 4.1. Suppose that Γ is a tsi Polish group, X is an analyticmetric space, Γ y X is Borel, R X ∆ is Borel for all open sets ∆ ⊆ Γ , (∆ k ) k ∈ N is a decreasing sequence of open subsets of Γ forming aneighborhood basis for Γ , and Γ k is the group generated by ∆ k . Thenexactly one of the following holds: (1) The action Γ y X is σ -lacunary. (2) There is a continuous injective homomorphism φ : 2 N × N → X from (∆(2 N ) k × E × ∆(2 N ) N ) k ∈ N to ( E X Γ k ) k ∈ N that is also a ho-momorphism from ∼ E N to ∼ E X Γ .Proof. Note that condition (2) is equivalent to the apparently weakerstatement in which φ is merely Borel, since we can always pass to adense G δ set C ⊆ N × N on which φ is continuous (see, for example,[Kec95, Theorem 8.38]), and then compose φ ↾ C with the map givenby Proposition 3.4. So by [BK96, Theorem 5.2.1], we can assume thatΓ y X is continuous.By passing to appropriate open subneighborhoods of 1 Γ , we can as-sume that ∆ k is symmetric and ∆ k +1 ⊆ ∆ k for all k ∈ N . As Γ is tsi,we can also assume that each ∆ k is conjugation invariant.Define G i,j = R X ∆ i \ R X ∆ j for all i, j ∈ N . By Propositions 2.1 and 2.3,condition (1) of Theorems 1.1 and 4.1 are equivalent. So by Theorem1.1, it is sufficient to show that condition (2) of Theorem 1.1 impliescondition (2) of Theorem 4.1. Towards this end, suppose that thereexist f : N → N and a continuous homomorphism φ : 2 N → X from( G ,k ) k ∈ N to ( G k,f ( k ) ) k ∈ N .Appeal to Proposition 3.1 to obtain a continuous homomorphism ψ : 2 N → N from ( G ,k ) k ∈ N to ( G ,f k (0) ) k ∈ N . By replacing φ with φ ◦ ψ ,we can assume that the former is a homomorphism from ( G ,k ) k ∈ N to( G f k (0) ,f k +1 (0) ) k ∈ N . By replacing (∆ k ) k ∈ N with (∆ f k (0) ) k ∈ N , and there-fore ( G i,j ) i,j ∈ N with ( G f k ( i ) ,f k ( j ) ) i,j ∈ N , we can assume that φ is a homo-morphism from ( G ,k ) k ∈ N to ( G k,k +1 ) k ∈ N .Fix an enumeration ( δ k ) k ∈ N of a countable dense subset of Γ, and forall k, ℓ ∈ N , let R k,ℓ denote the pullback of R Xδ ℓ ∆ k through φ . Proposi-tion 3.2 then yields functions g n : 2 The function φ is a homomorphism from ( G s ) s ∈ < N to ( E X Γ k | s | ) s ∈ < N .Proof. For each n ∈ N , let T n denote the graph on 2 n consisting of allpairs of the form ( s n − −| s | a ( i ) a s, s n − −| s | a (1 − i ) a s ), where i < s ∈ 2, and ℓ < m , then t ℓ a ( i ) a c and t ℓ +1 a ( i ) a c are G s n − −| uℓ | ,u ℓ -related, so φ ( t ℓ a ( i ) a c ) and φ ( t ℓ +1 a ( i ) a c ) are R Xγ n ( u ℓ )∆ kn -related, thus there is an elementof ( γ n ( u m − )∆ k n · · · γ n ( u )∆ k n ) − ∆ k n ( γ n ( u m − )∆ k n · · · γ n ( u )∆ k n ) sen-ding φ ( s a (0) a c ) to φ ( s a (1) a c ). As the conjugation invarianceand symmetry of ∆ k n ensure that this product is ∆ m +1 k n , it follows that φ ( s a (0) a c ) E X Γ kn φ ( s a (1) a c ).Set ℓ n = |{ m < n | k m = k n }| for all n ∈ N , and define ψ : 2 N × N → N by ψ ( c )( n ) = c ( k n , ℓ n ) for all c ∈ N × N and n ∈ N . Let D and E denotethe pullbacks of ∆( X ) and E X Γ through φ ◦ ψ . Lemma 4.3. Suppose that i ∈ N and F ∈ [ i × N ] < ℵ . Then E ismeager in D i,F .Proof. For all k ∈ N , let R k denote the pullback of R X ∆ k through φ ◦ ψ . As R i +2 R − i +2 ⊆ R i +1 , Proposition 3.3 ensures that R i +2 is mea-ger in D i,F . The Kuratowski-Ulam theorem therefore ensures that forcomeagerly-many c ∈ ( i × N ) \ F and all s ∈ F , comeagerly-many verticalsections of { ( d, d ′ ) ∈ ( N \ i ) × N × ( N \ i ) × N | c ∪ s ∪ d R i +2 c ∪ s ∪ d ′ } are mea-ger, so the fact that R − i +3 R i +3 ⊆ R i +2 implies that every vertical sectionof { ( d, d ′ ) ∈ ( N \ i ) × N × ( N \ i ) × N | c ∪ s ∪ d R i +3 c ∪ s ∪ d ′ } is meager. As ev-ery vertical section of { ( d, d ′ ) ∈ ( N \ i ) × N × ( N \ i ) × N | c ∪ s ∪ d E c ∪ s ∪ d ′ } is the union of countably-many such vertical sections, the Kuratowski-Ulam theorem yields that E is meager in D i,F .By composing φ ◦ ψ with the function obtained from applying Propo-sition 3.6 to D and E , we obtain the desired homomorphism.When Γ is non-archimedean, we obtain the following. Theorem 4.4. Suppose that Γ is a non-archimedean tsi Polish group, X is an analytic metric space, Γ y X is Borel, and E X Γ is Borel. Thenexactly one of the following holds: (1) The action Γ y X is σ -lacunary. (2) There is a continuous embedding π : 2 N × N → X of E N into E X Γ .Proof. By [BK96, Theorem 7.1.2], the orbit equivalence relation in-duced by every open subgroup of Γ is Borel. The fact that Γ is non-archimedean therefore implies that the orbit relation induced by everyopen subset of Γ is Borel.We can assume that Γ y X is continuous for exactly the same reasongiven at the beginning of the proof of Theorem 4.1.Fix a decreasing sequence (Γ k ) k ∈ N of normal subgroups of Γ forminga neighborhood basis for 1 Γ . In light of Theorem 4.1, we can assumethat there is a continuous injective homomorphism φ : 2 N × N → X from(∆(2 N ) k × E × ∆(2 N ) N ) k ∈ N to ( E X Γ k ) k ∈ N that is also a homomorphismfrom ∼ E N to ∼ E X Γ . But the continuity of Γ y X ensures that everysuch function is a reduction of E N to E X Γ . Acknowledgements. I would like to thank Alexander Kechris forpointing out the first sentence of the proof of Theorem 4.4. References [Bec98] Howard Becker, Polish group actions: dichotomies and generalized ele-mentary embeddings , J. Amer. Math. Soc. (1998), no. 2, 397–449.MR 1478843[BK96] Howard Becker and Alexander S. Kechris, The descriptive set theory ofPolish group actions , London Mathematical Society Lecture Note Series,vol. 232, Cambridge University Press, Cambridge, 1996. MR 1425877[GX14] Su Gao and Mingzhi Xuan, On non-Archimedean Polish groups withtwo-sided invariant metrics , Topology Appl. (2014), 343–353.MR 3132374[HK01] Greg Hjorth and Alexander S. Kechris, Recent developments in the theoryof Borel reducibility , Fund. Math. (2001), no. 1-2, 21–52, Dedicatedto the memory of Jerzy Lo´s. MR 1881047[Kec95] A.S. Kechris, Classical descriptive set theory , Graduate Texts in Math-ematics, vol. 156, Springer-Verlag, New York, 1995. MR 1321597(96e:03057)[Kle52] V. L. Klee, Jr., Invariant metrics in groups (solution of a problem ofBanach) , Proc. Amer. Math. Soc. (1952), 484–487. MR 0047250[KST99] A.S. Kechris, S. Solecki, and S. Todorcevic, Borel chromatic numbers ,Adv. Math. (1999), no. 1, 1–44. MR 1667145 (2000e:03132) SSENTIAL COUNTABILITY 15 Benjamin D. Miller, Kurt G¨odel Research Center for Mathemati-cal Logic, Universit¨at Wien, W¨ahringer Straße 25, 1090 Wien, Aus-tria E-mail address : [email protected] URL ::