Logic for metric structures and the number of universal sofic and hyperlinear groups

Using the model theory of metric structures, I give an alternative proof of the following result by Thomas: If the Continuum Hypothesis fails then there are power of the continuum many universal sofic groups up to isomorphism. This method is also applicable to universal hyperlinear groups, giving a positive answer to a question posed by Thomas.


Introduction
Sofic and hyperlinear groups are two classes of discrete groups that have received the attention of many mathematicians in different areas in the last ten years. It is known that the class of sofic groups is contained in the class of hyperlinear groups, but it is not known if this containment is proper, or whether the class of hyperlinear groups is equal to the class of all discrete groups. For a complete presentation of this topic, the reader is referred to [11]. In [3], Elek and Szabó proved that a countable group Γ is sofic if and only if it can be embedded in some (or, equivalently, every) ultraproduct of the symmetric groups, regarded as a biinvariant metric group with respect to the normalized Hamming distance. (See [1] for the definition of metric ultraproducts and an introduction to the logic for metric structures.) An analogous characterization holds for hyperlinear groups, where the symmetric groups are replaced with the finite rank unitary groups, endowed with the normalized distance induced by the Hilbert-Schmidt norm. In view of this characterization, metric ultraproducts of symmetric groups are said to be universal sofic groups, and metric ultraproducts of unitary groups are said to be universal hyperlinear groups. In [12], Thomas proved that, if the Continuum Hypothesis fails, then there are 2 c many metric ultraproducts of the symmetric groups up to (algebraic) isomorphism, where c denotes the cardinality of the continuum, and asked if the same statement holds for ultraproducts of the unitary groups. In this paper, I give a proof of Thomas' result, by means of the logic for metric structures, which also applies in the case of the unitary groups. From this, I deduce also the existence, under the failure of the Continuum Hypothesis, of 2 c many metric ultraproducts of the matrix algebras regarded as ranked regular rings, up d Sn (σ, τ ) = 1 n |{i ∈ {1, . . . , n} | σ (i) = τ (i) }| , called the normalized Hamming distance. The unitary group U n is endowed with the metric d Un (A, B) = A − B 2 2 √ n where · 2 denotes the Hilbert-Schmidt norm. Usually the factor 1 2 is omitted. It is introduced here only because, in the logic for bounded metric structures, for convenience all the metric spaces are supposed to have diameter at most 1. By the universal sofic and, respectively, hyperlinear groups, I will mean in the following the metric ultraproducts of the sequences of the symmetric and, respectively, unitary groups. Consider, for n ∈ N, the injective homomorphism σ → A σ from S n to U n defined by for i ∈ {1, 2, . . . , n}, where b 1 , . . . , b n is the canonical basis of C n , and observe that [11]).
In the rest of the paper, I will use the following notational conventions: If a, b are elements of a group G, then [a, b] denotes the element aba −1 b −1 of G. Upper case calligraphic letters such as U, V are reserved for ultrafilters over N. If (M n ) n∈N is a sequence of metric structures and U is an ultrafilter over N, the corresponding metric ultraproduct is denoted by U n M n , while the ultrapower of a metric structure M by U is denoted by M U . I will denote byx andȳ m-tuples of variables x 1 , . . . , x m and y 1 , . . . , y m . Every metric will be denoted by d. The context will make clear which metric I am referring to each time. The set of natural numbers N is supposed not to contain 0, and if r is a real number, then ⌈r⌉ denotes the smallest integer greater than or equal to r. For convenience, I suppose 0 to be a multiple of any natural number. Finally I will write, as usual, the acronym CH to stand for "Continuum Hypothesis".
I would like to thank my supervisor Ilijas Farah for his help and support, Samuel Coskey and Nicola Watson for their comments and suggestions, Lukasz Grabowski, Bradd Hart, Itaï Ben Yacoov, Ferenc Bencs, Louis-Philippe Thibault and Nigel Sequeira for many useful conversations.

The order property for symmetric and unitary groups
In [6], Theorem 6.1, aiming to count the number of ultrapowers of a C * -algebra or of a von Neumann algebra, Farah and Shelah isolate a condition ensuring that a sequence of metric structures has 2 c many ultraproducts up to isometric isomorphism, under the failure of CH.
In this section, I will consider a particular case of [6], Theorem 6.1, for biinvariant metric groups, and I will infer from that the following: Proposition 2.1. If CH fails and (k n ) n∈N is a strictly increasing sequence of natural numbers, then the sequences (S kn ) n∈N and (U kn ) n∈N have 2 c many ultraproducts up to isometric isomorphism.
In the following section, after introducing notations and definitions from [6], I will refine this result, showing that in this case there are in fact 2 c many ultraproducts up to algebraic isomorphism, under the failure of CH.
The following Proposition is a particular case of [6], Theorem 6.1, obtained by considering the language of bi-invariant metric groups and the formula Proposition 2.2. Let (k n ) n∈N be a strictly increasing sequence of natural numbers and (G n ) n∈N be a sequence of bi-invariant metric groups with uniformly bounded diameter. Suppose that, for some constant γ > 0 and every l ∈ N, for all but finitely many n ∈ N, G n contains sequences (g n,i ) Then, under the failure of CH, there are 2 c many pairwise non isometrically isomorphic metric ultraproducts of the sequence (G kn ) n∈N .
Thus, in order to prove Proposition 2.1, it is enough to show that the sequences of symmetric and unitary groups satisfy the hypothesis of Proposition 2.2. Lemma 2.4. If n, k, l ∈ N and r ∈ N∪{0} are such that n = 3 l k+r and 0 ≤ r < 3 l , then there are sequences (Σ n,i ) l i=1 and (T n,i ) l i=1 in S n such that, for 1 ≤ i, j ≤ l, Σ n,i and T n,i commute if i < j, while [Σ n,i , T n,j ] is the product of 3 l−1 k disjoint cycles of length 3 and, in particular, Proof. Consider the action of S 3 l on 1, . . . , 3 l × {1, . . . , k} defined by for every i ∈ 1, . . . , 3 l and j ∈ {1, . . . , k}. This defines an isometric embedding of S 3 l into S 3 l k . Moreover, letting S 3 l k act on the first 3 l k elements of {1, .., n} defines an algebraic embedding of S 3 l k into S n The composition Φ of these two embeddings is an algebraic embedding of S 3 l into S n . For 1 ≤ i ≤ l, define where σ l,i and τ l,i are the elements of S 3 l defined in Lemma 2.3. Then, if 1 ≤ i, j ≤ l, If i < j, [σ l,i , τ l,j ] is the identity and, hence, [Σ n,i , T n,j ] is the identity. If i ≥ j then [σ l,i , τ l,j ] is a product of 3 l−1 disjoint 3-cycles and, hence, [Σ n,i , T n,j ] is the product of 3 l−1 k disjoint 3-cycles.
Lemma 2.5. If n, k, l ∈ N and r ∈ N ∪ {0} are as in the statement of Lemma 2.4, then there are sequences where Σ n,i , T n,i ∈ S n are the permutations given by Lemma 2.4 Proposition is now an immediate consequence of Lemma 2.4 and Lemma 2.5, together with Proposition 2.2.

Non-isomorphic universal sofic and hyperlinear groups
In this section, I will prove the following strengthening of Proposition 2.1.
Theorem 3.1. If CH fails and (k n ) n∈N is an increasing sequence of natural numbers, then, up to algebraic isomorphism, there are 2 c many ultraproducts of both the sequence (S kn ) n∈N and the sequence (U kn ) n∈N . This result has already been proved by Thomas in [12] for permutation groups. Lukasz Grabowski pointed out to me that the case of unitary groups can be deduced from Proposition 8.3 in [6], using the facts that non-isomorphic type II von Neumann algebras have non-isomorphic unitary groups ( [7], Theorem 4) and that the unitary group of an ultraproduct of finite von Neumann algebras is the ultraproduct of the unitary groups ([8], Proposition 2.1). In the following, I will give a direct proof of this result by means of the logic for metric structures. This generalization of the usual discrete logic is suitable to deal with structures endowed with a metric. For an introduction to this topic, the reader is referred to [1]. Structures and formulas in the usual discrete logic can be considered particular cases of metric structures and formulas, where the distance is interpreted as the trivial discrete distance defined by d (x, y) = 1 iff x = y. Thus, definitions and theorems stated in the setting of the logic for metric structures subsume the analogous definitions and theorems for the usual discrete logic as a particular case. For the sake of simplicity, all the languages are henceforth supposed without relation symbols, apart from the metric. If M is a structure in such a language L, denote by M alg the L-structure obtained from M by replacing the metric on M by the trivial discrete metric.
I have now to introduce some notation and recall some results from [6]. If L is a language, ψ (x,ȳ) is an L-formula, ε ≥ 0 and M is an L-structure, the relation A chain in M k with respect to the relation ≺ ψ,ε will be called a (ψ, ε)-chain in M . The relation ≺ ψ,0 will be denoted by ≺ ψ and a (ψ, 0)-chain will be called a ψ-chain. If M is an L-structure and ϕ (x,ȳ) an L-formula, a ψ-chain C is called (ℵ 1 , ψ)-skeleton like if, for everyā ∈ M k there exists a countable Cā ⊂ C such that, for everyb,c ∈ C such that . The notion of ψ-chain and (ℵ 1 , ψ)-skeleton like ψ-chain in a discrete structure for a discrete formula ψ are obtained from the previous ones, as a particular case.
is an L-formula, I is a linear order of cardinality c and (M n ) n∈N is a sequence of L-structures such that, ∀n ∈ N, M n contains a ϕ-chain of length n, then there is an ultrafilter U over N such that U n M n contains an (ℵ 1 , ϕ)-skeleton like ϕ-chain of order type I.
The same fact for discrete structures and formulas can be inferred from this lemma as a particular case. The following definition, taken from [4], is of key importance for the proof of the main result. The assumption that a sequence (M n ) n∈N of L-structures has the order property is slightly weaker than the assumption on (M n ) n∈N in Lemma 3.2. Nonetheless, if a sequence (M n ) n∈N has the order property, then the same conclusion as in Lemma 3.2 holds, namely, for every linear order I of cardinality c, there is an ultrafilter U over N such that U n M n contains an (ℵ 1 , ϕ)-skeleton like ϕ-chain of order type I. This is easily seen via a suitable modification of the proof of Proposition 6.6 in [6].
The connection between the number of non-isomorphic ultraproducts and the (ℵ 1 , ϕ)-skeleton like ϕ-chains is given by the following lemma. It is proved in [6] (Proposition 3.14) in the setting of usual first order logic. As pointed out in Section 6.5 of the same paper, the proof can be easily adapted to the metric case.
Lemma 3.4. If CH fails, ϕ (x,ȳ) is an L-formula and K is a class of L-structures such that, for every linear order I of cardinality c, there is an element M of K such that M contains an (ℵ 1 , ϕ)-skeleton like ϕ-chain, then there are 2 c many pairwise non-isometrically isomorphic L-structures in K.
The following lemma is useful when, as in our case, one is interested in counting the number of metric ultraproducts up to algebraic isomorphism. Lemma 3.5. Suppose that ϕ (x,ȳ) is an L-formula and ψ (x,ȳ) is a discrete Lformula such that, for every L-structure M , for everyā,b ∈ M k , ψ ā,b holds in M alg if and only if ϕ M ā,b = 0. If M is an L-structure and C is an (ℵ 1 , ϕ)skeleton like ϕ-chain in M , then C is an (ℵ 1 , ψ)-skeleton like ψ-chain of the same order type in M alg .
Proof. The hypothesis implies that ≺ ψ in M k alg refines ≺ ϕ in M k . Thus, a ϕ-chain in M is also a ψ-chain in M alg of the same order type. Moreover, supposeā ∈ M k and Cā is as in the definition of (ℵ 1 , ϕ)-skeleton like. Ifb,c ∈ C are such that In the same way,  Proof. Since every subsequence of (M n ) n∈N has the order property witnessed by ϕ, it is no loss of generality to assume k n = n for every n ∈ N. By Lemma 3.2, for every linear order I of cardinality c, there is an ultrafilter U such that U n M n has an (ℵ 1 , ϕ)-skeleton like ϕ-chain C of order type I. By Lemma 3.5, C is also an (ℵ 1 , ψ)-skeleton like ψ-chain of the same order type in Observe that in this result 2 c is the maximum number possible, because it is the number of ultrafilters over N. The following result is an immediate consequence of Proposition 3.7 and Remark 3.6. are η-chains of length l in S n and U n respectively. An application of Corollary 3.8 concludes the proof of Theorem 3.1.

Ranked regular rings
If n ∈ N, denote by M n the algebra of n × n matrices over C and by rk the normalized rank on M n . Thus, if A ∈ M n , rk (A) is the rank of A divided by n. In [2], Elek considered metric ultraproducts of the matrix algebras over C with respect to the rank metric d (a, b) = rk (a − b) . If σ ∈ S n , denote, as before, by A σ the permutation matrix associated to σ, regarded as an element of M n . It is easily seen that where l is the number of possibly trivial cycles of σ. It can be deduced from Lemma 2.4 that, for any increasing sequence (k n ) n∈N of natural numbers, the sequence (M kn ) n∈N of matrix algebras endowed with the rank metric has the order property witnessed by the formula ϕ (x 1 , x 2 , y 1 , y 2 ) defined by min {3d (x 1 y 2 , y 2 x 1 ) , 1} .
The proof of this fact is left to the reader, being very similar to the proof of Lemma 2.5, using (4.1) instead of (1.1). The following Proposition, that answers a question of Elek, can now be obtained by direct application of Corollary 3.8.
Proposition 4.1. If CH fails, then for every increasing sequence (k n ) n∈N of natural numbers there are 2 c many metric ultraproducts of the sequence (M kn ) n∈N with respect to the rank metric whose multiplicative semigroups are pairwise nonisomorphic.
Proposition 4.1 can in fact be generalized to direct sequences of finite sums of matrix algebras obtained from a Bratteli diagram and a harmonic function as in [2], Section 3. Moreover, the same holds without change and with same proof if C is replaced by any other field or by any von Neumann regular ring R endowed with a rank function N . The easy details are left to the interested reader. An exhaustive treatment of von Neumann regular rings and ranked von Neumann regular rings can be found in [10].

The Σ 2 -theories of universal sofic and hyperlinear groups
If CH holds, then every universal sofic group and every universal hyperlinear group is saturated when regarded as a metric structure; and hence two such groups are isometrically isomorphic if and only if their metric theories coincide. (Again assuming CH, it is currently not known whether there exist algebraically nonisomorphic universal sofic groups or algebraically non-isomorphic universal hyperlinear groups; and in [12], Thomas asked whether all the universal sofic groups were elementarily equivalent when regarded as first-order structures.) In this section, I will prove the partial results that all universal sofic groups have the same metric Σ 2 -theories and that all universal hyperlinear groups have the same metric Σ 2 -theories. This is equivalent to the statement that for any Σ 2 formula ϕ in the language of metric groups, the sequences of real numbers given by evaluation of ϕ in the symmetric groups and, respectively, in the unitary groups, converge. Since a formula ϕ is Π 2 iff 1 − ϕ is Σ 2 , this implies that universal sofic, and respectively hyperlinear, groups have the same Π 2 theories as well.
The proof will make use of a general lemma, roughly asserting that if (M n ) n∈N is a sequence of structures such that, given m ∈ N, the elements of M n , for n large enough, can be arbitrarily well approximated by elements in the range of some approximate embedding of M m into M n , then for every Σ 2 sentence ϕ the sequence ϕ Mn n∈N converges. In order to precisely state and prove the lemma, I need to introduce the following terminology. A function ι : M → N between structures in a metric language is said to preserve all the function and relation symbols up to δ ≥ 0 if, for every n-ary function symbol f and a 1 , . . . , a n ∈ M , d N f N (ι (a 1 ) , .., ι (a n )) , ι f M (a 1 , . . . , a n ) ≤ δ and for every n-ary relation symbol R and a 1 , .., a n ∈ M , R N (ι (a 1 ) , . . . , ι (a n )) − R M (a 1 , . . . , a n ) ≤ δ.
Theorem 3.5 of [1] asserts that the interpretation of a formula ϕ in any structure is uniformly continuous in every variable, with uniform continuity modulus independent from the other variables and from the structure. It follows that for every ǫ > 0 there exists δ > 0 such that every embedding ι of a structure M into a structure N preserving all the function and relation symbols up to δ also preserves ϕ up to ǫ.
Observe that, if ψ (y, x) is a quantifier-free formula, then is a Σ 2 sentence. Conversely, by [1], Theorem 6.3, Proposition 6.6 and Proposition 6.9, the set of such sentences is dense in the set of all Σ 2 sentences. Thus, there is no loss of generality in considering only this type of Σ 2 sentences.
Lemma 5.1. Assume that (M n ) n∈N is a sequence of structures in a metric language L. Suppose that, ∀δ > 0, ∃m 0 ∈ N such that, ∀m ≥ m 0 , ∃k 0 ∈ N such that ∀k ≥ k 0 , ∀a ∈ M k there exists an embedding ι k m (possibly depending on a) of M m into M k satisfying the following properties: • ι k m preserves all the relation and function symbols up to δ • there isã ∈ M m such that d ι k m (ã) , a < δ. Then if ψ is a quantifier-free L-formula and ϕ is the L-formula inf x sup y1,..,,yn ψ (y 1 , . . . , y n , x) , then the sequence ϕ Mm m∈N converges. Proof. Fix ε > 0 and define δ > 0 such that any embedding preserving all the function and relation symbols up to δ preserves ψ up to ε, and moreover δ < ω (ε), where ω is a uniform continuity modulus for ψ in the last variable. Consider Finally, letting ε go to 0, one gets This concludes the proof.
I will now prove that the sequence (S n ) n∈N of symmetric groups satisfies the hypothesis of Lemma 5.1. The key observation is that a permutation σ ∈ S km belongs to the image of some isometric embedding of S m into S km if, for every l ≥ 1, the number l-cycles of σ is a multiple of k (here and in the following, I consider 0 to be multiple of any natural number). Thus, it is enough to prove that one can "chop up" any permutation σ ∈ S km , obtaining another permutationσ close to σ with the number of its l-cycles a multiple of k for every l ∈ N. Lemma 5.2 and Lemma 5.3 essentially show that any permutation is close to a permutation with only "small" cycles, while Lemma 5.4 shows that a permutation with only small cycles is close to one with the number of its l-cycles a multiple of k.
In order to simplify the discussion, I will introduce the following notation: If σ ∈ S n and l ≥ 1, define C l (σ) to be the set of cycles of σ of length l and w (σ) the greatest l such that C l (σ) is non-empty. In particular, C 1 (σ) is the set of fixed points of σ.
Proposition 5.5. There exists m 0 ∈ N such that, for every m ≥ m 0 and k ∈ N, if σ ∈ S km then there exists ρ ∈ S km such that C 1 (ρ) ⊃ C 1 (σ), w (ρ) ≤ ⌈ 3 √ m⌉, d (ρ, σ) ≤ 9 3 √ m and ρ = Φ (ρ) for someρ ∈ S m and isometric embedding Φ : S m → S km Proof. By Lemma 5.3, there is τ ∈ S km such that d (τ, σ) < 8 √ m⌉ and |C i (σ)| is a multiple of k for every i ∈ N. Thus, ρ = Φ (ρ) for someρ ∈ S m and isometric embedding Φ : S m → S km . Finally, If k, m ∈ N and 0 ≤ r < m, define ι to be the injective group homomorphism of S km into S km+r obtained by sending a permutation σ to the permutation that acts as σ on {1, 2, . . . , km} and fixes pointwise {km + 1, . . . , km + r}. Since ι preserves distances up to 1 − 1 m , and any element of S km+r is at distance at most 1 − 1 m from some element in the range of ι, it follows from Proposition 5.5 that the sequence (S n ) n∈N satisfies the hypothesis of Lemma 5.1. This concludes the proof of: in the language of bi-invariant metric groups, then the sequence ϕ Sn n∈N converges.

CONTINUOUS LOGIC AND UNIVERSAL SOFIC AND HYPERLINEAR GROUPS 13
The analogue of Proposition 5.5 in the case of unitary groups has been proved by von Neumann in [13], using the spectral theorem for normal matrices and an averaging argument on the eigenvalues. The precise statement is reported here for convenience of the reader. Observe that if W ∈ U km , then the function from U m to U km sending B to W (I k ⊗ B) W * is an isometric embedding. Here, ⊗ denotes the usual tensor product of matrices.
Proposition 5.7. If ε > 0, there exists m 0 ∈ N such that, for every k ∈ N and m ≥ n 0 , if A ∈ M km is a normal matrix with operator norm at most 1, there exists B ∈ M m of operator norm at most 1 and W ∈ U km such that A − W (I k ⊗ B) W * 2 < ε, where · 2 is the normalized Hilbert-Schmidt norm. Moreover, if A is Hermitian (resp. unitary), B can be chosen Hermitian (resp. unitary).
In order to show that the sequence of unitary groups satisfies the hypothesis of Lemma 5.1 it remains only to show that, if k, m ∈ N and 0 ≤ r < k, then there is an injective group homomorphism ι of U km into U km+r that almost preserves the metric and such that any element of U km+r is close to some element in the range of ι. This is done in the following lemma.
Lemma 5.8. If k, m ∈ N and 0 ≤ r < m, then the function ι from U km to U km+r sending A to A 0 0 I r (where I r is the r × r identity matrix) is an injective group homomorphism that preserves the metric up to 1 k . Moreover, any element of U km+r is at distance at most Suppose now that C ∈ U n and define A to be the element of M km such that A i,j = C i,j for 1 ≤ i, j ≤ km. It is easy to see that, since U is unitary, By [9], Corollary 1, there exists B ∈ U km such that Thus, This concludes the proof of: in the language of bi-invariant metric groups, then the sequence ϕ Un n∈N converges.