aa r X i v : . [ m a t h . L O ] J u l Metric groups, unitary representations andcontinuous logic
Aleksander Ivanov (Iwanow)July 9, 2020
Abstract
We describe how properties of metric groups and of unitary representa-tions of metric groups can be presented in continuous logic. In particularwe find L ω ω -axiomatization of amenability. We also show that in thecase of locally compact groups some uniform version of the negation ofKazhdan’s property (T) can be viewed as a union of first-order axiom-atizable classes. We will see when these properties are preserved undertaking elementary substructures. In this paper we study the behavior of amenability and Kazhdan’s property (T) under logical constructions. We view these tasks as a part of investigationsof properties of basic classes of topological groups appeared in measurable andgeometric group theory, see [13], [14], [18]. The fact that some logical construc-tions, for example ultraproducts, have become common in group theory, givesadditional flavour for this topic. We concentrate on properties of metric groupswhich can be expressed in continuous logic [2].Since we want to make the paper available for non-logicians, in Section 2 webriefly remind the reader some preliminaries of continuous logic. In Section 3we apply it to amenability. In Section 4 we consider unitary representations oflocally compact groups.
We fix a countable continuous signature L = { d, R , ..., R k , ..., F , ..., F l , ... } . Let us recall that a metric L -structure is a complete metric space ( M, d ) with d bounded by 1, along with a family of uniformly continuous operations F i on M R i , i.e. uniformly continuous maps from appropriate M k i to [0 , R i a continuitymodulus γ i is assigned so that when d ( x j , x ′ j ) < γ i ( ε ) with 1 ≤ j ≤ k i thecorresponding predicate of M satisfies | R i ( x , ..., x j , ..., x k i ) − R i ( x , ..., x ′ j , ..., x k i ) | < ε. It happens very often that γ i coincides with id . In this case we do not mentionthe appropriate modulus. We also fix continuity moduli for functional sym-bols. Each classical first-order structure can be considered as a complete metricstructure with the discrete { , } -metric.By completeness, continuous substructures of a continuous structure arealways closed subsets.Atomic formulas are the expressions of the form R i ( t , ..., t r ), d ( t , t ), where t i are terms (built from functional L -symbols). In metric structures they cantake any value from [0 , Statements concerning metric structures are usuallyformulated in the form φ = 0(called an L - condition ), where φ is a formula , i.e. an expression built from 0,1and atomic formulas by applications of the following functions: x/ x ˙ − y = max ( x − y,
0) , min ( x, y ) , max ( x, y ) , | x − y | , ¬ ( x ) = 1 − x , x ˙+ y = min ( x + y,
1) , sup x and inf x . A theory is a set of L -conditions without free variables (here sup x and inf x playthe role of quantifiers).It is worth noting that any formula is a γ -uniformly continuous functionfrom the appropriate power of M to [0 , γ is the minimum of continuitymoduli of L -symbols appearing in the formula.The condition that the metric is bounded by 1 is not necessary. It is of-ten assumed that d is bounded by some rational number d . In this case the(truncated) functions above are appropriately modified.We sometimes replace conditions of the form φ ˙ − ε = 0 where ε ∈ [0 , d ] bymore convenient expressions φ ≤ ε .In several places of the paper we use continuous L ω ω -logic. It extends thefirst-order logic by new connectives applied to countable families of formulas : W is the infinitary min and V corresponds to the infinitary max . When we applythese connectives we only demand that the formulas of the family all obey thesame continuity modulus, see [3]. Below we always assume that our metric groups are continuous structures withrespect to bi-invariant metrics (see [2]). This exactly means that (
G, d ) is acomplete metric space and d is bi-invariant. Note that the continuous logicapproach takes weaker assumptions on d . Along with completeness it is only2ssumed that the operations of a structure are uniformly continuous with respectto d . Thus it is worth noting here that • any group which is a continuous structure has an equivalent bi-invariantmetric.Indeed assuming that ( G, d ) is a continuous metric group which is not discreteone can apply the following function: d ∗ ( x, y ) = sup u,v d ( u · x · v, u · y · v ) . See Lemma 2 and Proposition 4 of [15] for further discussions concerning thisobservation.
In Section 3 we apply the recent paper [19] for L ω ω -axiomatization of amenability/non-amenability of metric groups. The case of property (T) looks slightly morecomplicated, because unbounded metric spaces are involved in the definition.Typically unbounded metric spaces are considered in continuous logic asmany-sorted structures of n -balls of a fixed point of the space ( n ∈ ω ). Section15 of [2] contains nice examples of such structures. If the action of a boundedmetric group G is isometric and preserves these balls we may consider the actionas a sequence of binary operations where the first argument corresponds to G .In such a situation one just fixes a sequence of continuity moduli for G (for each n -ball). We will see in Section 4 that this approach works sufficiently well forthe negation of property ( T ) (non-( T )) in the class of locally compact groups.It is well-known that a locally compact group with property (T) is amenableif and only if it is compact. Thus it is natural to consider these propertiestogether. Actions of metric groups which can be analyzed by tools of continuous logicmust be uniformly continuous for each sort appearing in the presentation of thespace by metric balls. This slightly restricts the field of applications.
We treat a Hilbert space over R exactly as in Section 15 of [2]. We identify itwith a many-sorted metric structure( { B n } n ∈ ω , , { I mn } m 2) For every θ ∈ (0 , , every finite subset E ⊆ G , and every identityneighbourhood U , there is a finite non-empty subset F ⊆ G such that ∀ g ∈ E ( µ ( F, gF, U ) ≥ θ | F | ) . (3) There exists θ ∈ (0 , such that for every finite subset E ⊆ G , and every identity neighbourhood U , there is a finite non-emptysubset F ⊆ G such that ∀ g ∈ E ( µ ( F, gF, U ) ≥ θ | F | ) . It is worth noting here that when an open neighbourhood V contains U the num-ber µ ( F, gF, U ) does not exceed µ ( F, gF, V ). In particular in the formulationabove we may consider neighbourhoods U from a fixed base of identity neigh-bourhoods. For example in the case of a continuous structure ( G, d ) with aninvariant d we may take all U in the form of metric balls B Given k ∈ N and rational numbers q, θ ∈ (0 , there is a quantifierfree formula φ k,q,θ (¯ x, y ) depending on variables x , . . . , x k and y such that in thestructure ( G, d ) the -statement φ k,q,θ (¯ x, y ) ≤ is equivalent to the conditionthat x , . . . , x k form a set F with µ ( F, yF, B q ) ≥ θ | F | .Moreover the identity function is a continuity modulus of y in φ k,q,θ (¯ x, y ) .Proof. To guarantee the inequality µ ( F, gF, B q ) ≥ θ | F | for an F = { f , . . . , f k } we only need to demand that for every S ⊆ F the following inequality holds: | S | − k + θ · k ≤ | N R ( S ) | , where N R ( S ) is defined with respect to ( F, gF ) and U = B q .To satisfy this inequality we will use the observation that when S ′ ⊆ gF and ρ is a function S ′ → S such that max { d ( gf, ρ ( gf )) : gf ∈ S ′ } ≤ q then | S ′ | ≤ | N R ( S )) | . Let us assume that S corresponds to some tuple x i , . . . , x i l ofelements of { x , . . . , x k } , the subset S ′ corresponds to some tuple of terms from { yx , . . . , yx k } (recovered by ρ − ) and let dist S,S ′ ,ρ ( x , . . . x k , y ) = max { d ( yx i , ρ ( yx i )) : yx i ∈ S ′ } (a max -formula of continuous logic). Then the statement formalizing | S ′ | ≤| N R ( S )) | can be expressed that the formula dist S,S ′ ,ρ ( x , . . . x k , y ) takes value ≤ q with respect to the realization of ¯ x by the tuple f , . . . f k and y by g .5hus the following formula φ k,q,θ (¯ x, y ) satisfies the statement of the lemma: max S ⊆{ x ,...,x k } min { dist S,S ′ ,ρ ( x , . . . x k , y ) : S ′ ⊆ { yx , . . . , yx k } , ρ : S ′ → S , | S | − k + θ · k ≤ | S ′ |} ˙ − q. To see the last statement of the lemma it suffices to notice that the identityfunction is a continuity modulus of y in each dist S,S ′ ,ρ ( x , . . . x k , y ). The lat-ter follows from the definition of dist S,S ′ ,ρ ( x , . . . x k , y ) and the fact that d isinvariant and satisfies the triangle inequality. Theorem 3.2. The class of all amenable groups which are continuous structureswith invariant metrics, is axiomatizable by all L ω ω -statements of the followingform: sup y ...y l _ { inf x ...x k max { φ k,q,θ (¯ x, y i ) : 1 ≤ i ≤ l } : k ∈ ω } ≤ , where θ, q ∈ Q ∩ (0 , and l ∈ ω. In particular every first order elementary substructure of a continuous structurewhich is an amenable group is also an amenable group.Proof. The tuple y , . . . , y l consists of all free variables of the formula inf x ...x k max { φ k,q,θ (¯ x, y i ) : 1 ≤ i ≤ l } . By the last statement of Lemma 3.1 the identity function is a continuity modulusof each y i in this formula. This implies the same statement concerning theinfinite disjunction in the formulation. Thus the formula in the formulationbelongs to L ω ω .By Theorem 4.5 of [19] and the discussion after the formulation of thattheorem above we see that all amenable groups satisfy the statements in theformulation.Let us prove the contrary direction. Let ( G, d ) satisfy the axioms from theformulation. We want to apply condition (2) of Theorem 4.5 of [19] in the caseof balls B q . Fix θ, q and E as in the formulation so that E = { g , . . . , g l } .Choose q ′ ∈ Q ∩ (0 , 1) so that q ′ < q . Since ( G, d ) satisfies sup y ...y l _ { inf x ...x k max { φ k,q ′ ,θ (¯ x, y i ) : 1 ≤ i ≤ l } : k ∈ ω } ≤ , we find f , . . . , f k ∈ G so that max { φ k,q ′ ,θ ( ¯ f , g i ) : 1 ≤ i ≤ l } < q − q ′ . By the definition of the formula φ k,q ′ ,θ ( ¯ f , g i ) we see that condition (2) of Theo-rem 4.5 of [19] holds for θ , U = B q and E . This proves that ( G, d ) is amenable.To see the last statement of the theorem assume that ( G, d ) is amenableand ( G , d ) (cid:22) ( G, d ). We repeat the argument of the previous paragraph for6 ⊆ G . Then having found f , . . . , f k ∈ G as above we can apply the definitionof an elementary substructure in order to obtain f ′ , . . . , f ′ k ∈ G so that max { φ k,q ′ ,θ ( ¯ f ′ , g i ) : 1 ≤ i ≤ l } < q − q ′ . The rest is clear.To have a similar theorem for non-amenability we need the following lemma. Lemma 3.3. Given k ∈ N and rational numbers q, θ ∈ (0 , there is a quantifierfree formula φ − k,q,θ (¯ x, y ) depending on variables x , . . . , x k and y such that in thestructure ( G, d ) the -statement φ k,q,θ (¯ x, y ) ≤ is equivalent to the conditionthat x , . . . , x k form a set F with µ ( F, yF, B G, d ) is not amenable.Then applying Theorem 4.5 (2) of [19] we find θ ∈ (0 , E = { g , . . . , g l } ⊆ G , and an identity neighbourhood U = B It is well-known that a locally compact group with property (T) of Kazhdanis amenable if and only if it is compact. Thus axiomatization of property (T) (non- (T) ) is natural in the context of axiomatization of amenability (at least forlocally compact groups). In this section we apply continuous logic to (T) /non- (T) . Our results are partial. On the one hand they are restricted to the classof locally compact groups and on the other one they mainly concern propertynon- (T) . Let a topological group G have a continuous unitary representation on a complexHilbert space H . A closed subset Q ⊂ G has an ε -invariant unit vector in H if there exists v ∈ H such that sup x ∈ Q k x ◦ v − v k < ε and k v k = 1 . A closed subset Q of the group G is called a Kazhdan set if there is ε > G on a Hilbert spacewhere Q has an ε -invariant unit vector there is a non-zero G -invariant vector.The following statement is Proposition 1.1.4 from [1].Let G be a topological group. The pair ( G, √ 2) is Kazhdan pair, i.e.if a unitary representation of G has a √ G has a non-zero invariant vector.If the group G has a compact Kazhdan subset then it is said that G has prop-erty ( T ) of Kazhdan .Proposition 1.2.1 of [1] states that the group G has property ( T ) of Kazhdanif and only if any unitary representation of G which weakly contains the unitrepresentation of G in C has a fixed unit vector.By Corollary F.1.5 of [1] the property that the unit representation of G in C is weakly contained in a unitary representation π of G (this is denoted by1 G ≺ π ) is equivalent to the property that for every compact subset Q of G andevery ε > Q has an ε -invariant unit vector with respect to π .The following example shows that in the first-order logic property ( T ) is notelementary: there are two groups G and G which satisfy the same sentencesof the first-order logic but G | = ( T ) and G = ( T ). Example. Let n > 2. According Example 1.7.4 of [1] the group SL n ( Z )has property ( T ). Let G be a countable elementary extension of SL n ( Z ) whichis not finitely generated. Then by Theorem 1.3.1 of [1] the group G does nothave ( T ).The main result of this section, Theorem 4.4, shows that in the context ofcontinuous logic the class of unitary representations of locally compact groupswith property non- (T) can be viewed as the union of axiomatizable classes.9 .2 Unitary representations in continuous logic We apply methods announced in the introduction. In order to treat axiomatiz-ability question in the class of locally compact groups satisfying some uniformversion of property non-( T ) we need the preliminaries of continuous model the-ory of Hilbert spaces from Section 2.5. Moreover since we want to considerunitary representations of metric groups G in continuous logic we should fixcontinuity moduli for the corresponding binary functions G × B n → B n inducedby G -actions on metric balls of the corresponding Hilbert space. Remark 4.1. Continuous unitary actions of G on B obviously determine theirextensions to S { B i : i > } : g ( r · x ) = r · g ( x ) where x ∈ B and r · x ∈ B n . Thus a continuity modulus, say F , for the corresponding function G × B → B can be considered as a family of continuity moduli for G × B i → B i as follows: F i ( ε ) = F ( εi ) . Using this observation we simplify the approach by considering only continuitymoduli for G × B → B . When we fix such F we call the correspondingcontinuous unitary action of G an F - continuous action.We now define a uniformly continuous versions of the notion of a Kazhdanset. Definition 4.2. Let G be a metric group of diameter ≤ d, · , − , F be a continuity modulus for the G -variable of continuous functions G × B → B .We call a closed subset Q of the group G an F - Kazhdan set if there is ε with the following property: every F -continuous unitary representation of G ona Hilbert space with ( Q, ε )-invariant unit vectors also has a non-zero invariantvector.It is clear that for any continuity modulus F a subset Q ⊂ G is F -Kazhdanif it is Kazhdan. We will say that G has property F - non-(T) if G does nothave a compact F -Kazhdan subset.To study such actions in continuous logic let us consider a class of many-sorted continuous metric structures which consist of groups G together withmetric structures of complex Hilbert spaces( d, · , − , ∪ ( { B n } n ∈ ω , , { I mn } m Let F be a continuity modulus for the G -variable of continuousfunctions G × B → B .(a) In the class of all unitary F -representations of locally compact metricgroups the condition of weak containment of the unit representation G coin-cides with the condition that the corresponding structure A ( G, H ) belongs to S {K δ ( F ) : δ ∈ (0 , ∩ Q } .(b) In the class of all unitary F -representations of locally compact metricgroups the condition of witnessing F -non- (T) corresponds to a union of axiom-atizable classes of structures of the form A ( G, H ) .Proof. (a) Let G be a locally compact metric group and let the ball K δ = { g ∈ G : d ( g, ≤ δ } ⊆ G be compact. If a unitary F -representation of G weaklycontains the unit representation 1 G , then considering it as a structure A ( G, H )we see that this structure belongs to K δ ( F ).On the other hand if some structure of the form A ( G, H ) belongs to K ε ( F ),then assuming that ε ≤ δ we easily see that the corresponding representationweakly contains 1 G . If δ < ε , then K ε may be non-compact. However since K δ ⊆ K ε any compact subset of G belongs to a finite union of sets of the form xK ε . Thus the axioms of K ε ( F ) state that the corresponding structure A ( G, H )defines a representation weakly containing 1 G .(b) The condition sup v ∈ B inf x ∈ G (1 ˙ − ( k x ◦ v − v k + | − k v k | )) ≤ NIV ) obviuosly implies that G does not have invariant unit vectors.For the contrary direction we use the fact mentioned in the introduction ofSection 4 that the absence of G -invariant unit vectors implies that G does nothave √ v ∈ B there is x ∈ G such that √ k v k≤k x ◦ v − v k . NIV .Adding NIV to each K δ ( F ) we obtain axiomatizable classes as in the state-ment of (b). Below we call it K NIV δ ( F ). (I) It is clear that the classes S {K δ ( F ) : δ ∈ (0 , ∩ Q } and S {K NIV δ ( F ) : δ ∈ (0 , ∩ Q } can be considered without the restriction of local compactness.However it does not look likely that then they axiomatize weak containment ofthe unit representation 1 G or witnessing non- (T) . (II) In spite of axiomatizability issues in our paper it is still not clear how largethe class of locally compact non-compact groups with bi-invariant merics andproperty (T) . This is especially interesting in the case of connected groups,since some standard Lie group examples do not admit compatible bi-invariantmetrics. (III) Using the method of Proposition 1.2.1 from [1] one can show that if forevery compact subset Q of a locally compact metric group ( G, d ) and every ε > G to a structure from K GH ( F ) with a ( Q, ε )-invariantunit vector but without non-zero invariant vectors, then G has a unitary F -representation which belongs to K NIV δ ( F ) for appropriate δ ∈ (0 , ∩ Q . (IV) Since the class of locally compact metric groups is not axiomatizable , the subclasses of K δ ( F ) and K NIV δ ( F ) appearing in Theorem 4.4 are only relatively axiomatizable . On the other hand they have some nice properties ofaxiomatizable classes. For example the following statement holds. Proposition 5.1. Any elementary substructure of any A ( G, H ) ∈ S {K NIV δ ( F ) : δ ∈ (0 , ∩ Q } with a locally compact G also belongs to S {K NIV δ ( F ) : δ ∈ (0 , ∩ Q } and is of the form A ( G , H ) , where G (cid:22) G , H (cid:22) H and G islocally compact. In the proof we use an additional tool from model theory. Let M be acontinuous metric structure. A tuple ¯ a from M n is algebraic in M over A ifthere is a compact subset C ⊆ M n such that ¯ a ∈ C and the distance predicate dist (¯ x, C ) is definable (in the sense of continuous logic, [2]) in M over A . Let acl ( A ) be the set of all elements algebraic over A . In continuous logic the conceptof algebraicity is parallel to that in traditional model theory (see Section 10 of[2]). Proof. Let M ∈ S {K NIV δ ( F ) : δ ∈ (0 , ∩ Q } and G be the group sort of M .Choose δ > δ -ball K = { g ∈ G : d ( g, ≤ δ } in G is compact. Notethat since the condition d ( g, ≤ δ defines a totally bounded complete subsetin any elementary extension of G , the set K is a definable subset of acl ( ∅ ).Let M (cid:22) M and G be the sort of M corresponding to G . It remains toverify that for any compact subset D ⊂ G and any ε > M always has a ( D, ε )-invariant unit vector. To see this note that since G ≺ G and an ultraproduct of compact metric groups is not necessarily locally compact is compact and algebraic, the ball { g ∈ G : d ( g, ≤ δ } ⊂ G is a compactneighborhood of 1 which coincides with K . In particular D is contained in afinite union of sets of the form gK . The rest follows from the conditions that M ∈ K δ ( F ) and G ≺ G .We do not know if the statement of this proposition holds without the as-sumtion that G locally compact. (V) In Section 8.4 of [18] it is proved that if Γ is a discrete group with property (T) , then the direct power Γ ω also has property (T) as a topological group. Onthe other hand the topological group Γ ω is a continuous metric group under theobvious metric. There is also a certain class of ’trivial examples’ of non-locallycompact groups with bi-invariant metrics that have property (T) . Namely, thereare abelian metrizable groups that admit no non-trivial unitary representations,i.e. satisfying property (T) . Such an example can be found in [6]. Since it isextremely amenable it does not admit non-trivial unitary representations. Theauthor does not know other non-locally compact groups which are continuousmetric groups with property (T) . In particular are there non-trivial connectedexamples? This remark originally appeared in a discussion with Michal Douchaand then was extended by the referee. (VI) The author thinks that the following question is basic in this topic. Let ( G, d ) be a metric group which is a continuous structure. Assumethat property (T) holds in G . Does every elementary substructureof ( G, d ) satisfy (T) ? According the previous remark it looks reasonable to start with the discretecase: Does an elementary substructure of a discrete group with property (T) also have property (T) ? It is natural to consider this question in the case of linear groups, where property (T) and elementary equivalence are actively studied, see [4] and [7]. (VII) Property FH states that every action of G by affine isometries on aHilbert space has a fixed point. It is equivalent to property (T) for σ -compactlocally compact groups. Axiomatization of FH is studied in arXiv paper [16].Since in this case unbounded actions appear, the approach is different there. (VIII) One of the definitions of non-amenability says that a topological groupis non-amenable if there is a locally convex topological vector space V and acontinuous affine representation of G on V such that some non-empty invariantconvex compact subset K of V does not contain a G -fixed point ([1], TheoremG.1.7). If we restrict ourselves just by linear representations on normed/metricvector spaces we obtain a property which is stronger than non-amenability. Wecall it strong non-FP . The paper [16] contains some results showing that theapproach to non- (T) presented in Sections 4 and 5 can be applied to strongnon-FP too. We would also mention that this arXiv paper also considers theclass of groups which are not extremely amenable in some uniform way.13 cknowledgments The research was partially supported by Polish National Science Centre grantDEC2011/01/B/ST1/01406.The author is grateful to the referee for helpful remarks. For example somereferee’s observations are included into Comments (I) - (VIII) .Section 4 of this paper is a version of Section 2 of [16] (which was split).The author is grateful to the referee of [16] for an advice concerning the proofof Theorem 4.4. References [1] B. Bekka, P. de la Harpe, P. Valette, Kazhdan’s Property (T) . New Math-ematical Monographs, 11, Cambridge University Press, Cambridge, 2008.[2] I. Ben Yaacov, A. Berenstein, W. Henson, A. Usvyatsov, Model theoryfor metric structures , in: Model theory with Applications to Algebra andAnalysis, v.2 , Z. Chatzidakis, H.D. Macpherson, A. Pillay, A. Wilkie(eds.), London Math. Soc. 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q and E = { g , . . . , g l } so that ( G, d ) satisfies ^ { sup x ...x k min { φ − k,q,θ (¯ x, g i ) : 1 ≤ i ≤ l } : k ∈ ω } ≤ ε. This obviously means that ^ { sup x ...x k min { φ − k,q − ε,θ (¯ x, g i ) : 1 ≤ i ≤ l } : k ∈ ω } ≤ . By the definition of the formula φ − k,q − ε,θ (¯ x, g i ) we see that condition (3) ofTheorem 4.5 of [19] does not hold for θ , U = B