Sofic representations of amenable groups

Using probabilistic methods, Collins and Dykema proved that the free product of two sofic groups amalgamated over a monotileably amenable subgroup is sofic as well. We show that the restriction is unnecessary; the free product of two sofic groups amalgamated over an arbitrary amenable subgroup is sofic. We also prove a group theoretical analogue of a result of Kenley Jung. A finitely generated group is amenable if and only if it has only one sofic representation up to conjugacy equivalence.


Introduction
First, let us recall the notion of sofic groups (see [8] for a survey) . Let Γ be a finitely generated group with a symmetric generating system S. Let G be a finite graph, such that each directed edge of G is labeled by an element of S. We say that G is an r-approximation of the Cayley-graph of Γ, Cay(Γ, S) if there exists a subset W ⊆ V (G) such that • |W | > (1 − 1 r )|V (G)| and • if p ∈ W , then the r-neighborhood of p is rooted isomorphic to the rneighborhood of a vertex of the Cayley-graph of Γ (as edge labeled graphs).
The group Γ is sofic if for any r ≥ 1 there exist r-approximations of Cay(Γ, S) by finite graphs. We call a not necessarily finitely generated group sofic if all of its finitely generated subgroups are sofic. Amenable and residually finite groups are sofic. Free products, direct products, inverse and direct limits of sofic groups are all sofic groups as well. Amenable extensions of sofic groups are also sofic. It is important to note that no non-sofic group is known at this moment. Several conjectures about groups, such as the Gottschalk's Conjecture, Kaplansky's Direct Finiteness Conjecture, the Determinant Conjecture and the Connes Embedding Conjecture hold for sofic groups. Using probabilistic arguments, Collins and Dykema [1] proved that the free products of sofic groups amalgamated over monotileably amenable groups are sofic. We show that the restriction is not necessary.
Theorem 1. Free products of sofic groups amalgamated over arbitrary amenable groups are sofic.
In [6], Kenley Jung gave a characterization of amenable von Neumann algebras among "embeddable" von Neumann algebras. He proved that an embeddable von Neumann algebra M is amenable if and only if any two embeddings of M into R ω are conjugate by a unitary element. Before stating our result, let us briefly recall an equivalent definition of soficity. Let ω be a non-principal ultrafilter on the natural numbers and lim ω : l ∞ (N) → R be the associated ultralimit. Consider the product of finite symmetric groups ∞ n=1 S n . If g ∈ S n then let the rank of g defined as rk(g) : n , where f ix(g) is the number of elements in the set {1, 2, . . . , n} fixed by the permutation g. Let H ⊂ ∞ n=1 S n be defined as follows.
Theorem 2. A finitely generated group Γ is amenable if and only if any two faithful embeddings of Γ into Σ are conjugate.
Note added: Liviu Paunescu informed us that he also has a proof of Theorem 1. using a different argument. [7] 2 Sofic approximations of amenable groups Definition 2.1. Let E(A) denote the set of edges in a colored graph A. We say that two colored graphs A and B are r-isomorphic for some r > 0 if there are subgraphs A ′ ⊆ A and B ′ ⊆ B such that | and A ′ isomorphic to B ′ (as colored graphs). The isomorphism between A ′ and B ′ is called an r-isomorphism. Lemma 2.2. Let A, B and C be colored graphs, A is 2r-isomorphic to B and B is 2r-isomorphic to C. Then A is r-isomorphic to C. If A is a 2r-approximation of a Cayley graph and every vertex of A has degree at most d then there is a bound R appr (r, d) such that every colored graph R appr (r, d)-isomorphic to A is an r-approximation of the Cayley graph.
Proof. Clear from the definition. Definition 2.3. Let T = T 1 , T 2 , . . . T m be a finite sequence of colored graphs. Their linear combination with coefficient vector α = (α 1 , α 2 , . . . α m ) ∈ N m , denoted by α · T , is the disjoint union of α 1 copies of T 1 , α 2 copies of T 2 , . . . and α m copies of T m . As a special case, if m = 1 then we talk about the integer multiples of T 1 .
Proof. Let E be the maximum number of edges in T i and let t be the largest integer such that tα ≤ β . Clearly M (r) ≤ t and α ≥ 1, hence hence each coordinate of β differs by at most 2 M(r) α t from the corresponding coordinate of tα. Therefore β · T can be obtained from (tα) · T by adding or deleting at most 2 M(r) α t copies of each T i , and the total number of edges involved in this operation is at most 2mE M(r) α t. On the other hand, the sum of the coordinates of tα is at least tα (triangle inequality), hence the graph tα · T has at least α t edges, and similarly, the graph β · T has at least β ≥ α t edges. If we choose M (r) ≥ 2mEr then we obtain that β · T is r-isomorphic to (tα) · T , which is indeed an integer multiple of α · T . Definition 2.6. Let Γ be a finitely generated group, S ⊆ Γ a finite symmetric generating set. A Følner sequence in the Cayley graph Cay(Γ, S) is a sequence F = {F 1 , F 2 , . . . } of spanned subgraphs such that for all r > 0 all but finitely many of the F n are r-approximations. The group Γ is amenable if Cay(Γ, S) has a Følner sequence. Note, that this property depends only on the group, i.e. any other Cayley graph of Γ has a Følner sequence. More generally, an arbitrary group is amenable if each of its finitely generated subgroups is amenable. [2,Theorem 2.]] Let Γ be a finitely generated amenable group, S ⊆ Γ a finite symmetric generating set and F a Følner sequence in Cay(Γ, S). Then for each integer r > 0 there is an integer R 0 (r) > 0 (which depends also on F ) and a finite subsequence T ⊂ F such that each R 0 (r)approximation of Cay(Γ, S) is r-equivalent to some linear combination of T .
Proposition 2.8. Let Γ be a finitely generated amenable group, S ⊆ Γ a finite symmetric generating set and F a Følner sequence in Cay(Γ, S). Then for each integer r > 0 there is an integer R amenable (r) (which depends also on F ) and a finite subsequence T ⊂ F of say m members and a lattice vector 0 = α ∈ N m with the following property. Each member of T is an 2r-approximation of Cay(Γ, S), and each R amenable (r)-approximation of Cay(Γ, S) is 2r-equivalent to an integer multiple of the (fixed) linear combination α · T .
Proof. By discarding a few elements from F we may assume that the remaining elements are all 2r-approximations of Cay(Γ, S). We apply Proposition 2.7 with parameter 8r, we obtain a subsequence T ⊂ F of say m members such that every R 0 (8r)-approximation of Cay(Γ, S) is 8r-isomorphic to some linear combination of T . Now we discard those elements of F that are not R 0 (8r)-approximation of Cay(Γ, S). Note, that T is no longer a subsequence of the remaining F .
For each member F ∈ F we choose a lattice vector 0 = β F ∈ N m such that β F · T is 8r-isomorphic to F . Now σ(β F ) F ∈ F } is a sequence of unit vectors, so it has an accumulation point u ∈ R m . We apply Lemma 2.5 to T with parameter 8r and obtain the bound M (8r). Let us fix a member H ∈ F such that σ(β H ) − u < 1 2M(8r) , and then replace F by the subsequence of those F ∈ F which satisfy σ(β F ) − u < 1 2M(8r) and β F ≥ M (8r) σ(β H ) . The remaining sequence F is still a Følner sequence and for each F ∈ F we have σ(β F ) − σ(β H ) < 1 M(8r) , hence β F · T is 8r-isomorphic to an integer multiple of β H · T by Lemma 2.5.
Applying again Lemma 2.7 with parameter 8r we obtain new finite subsequence {Q 1 , Q 2 , . . . Q n } of our modified F . Let X be any R 0 (8)-approximation of Cay(Γ, S). Then X is 8r-isomorphic to a linear combination of the Q i , each Q i is 8r-isomorphic to the corresponding β Qi · T , and in turn each β Qi · T is 8r-isomorphic to an integer multiple of β H · T . Applying Lemma 2.2 twice we obtain that X is 2r-isomorphic to an integer multiple of β H · T . The Lemma holds with R amenable (r) = R 0 (8r).

Amalgamated products
Definition 3.1. Let α and β be partitions of a finite set S. The incidence graph of α and β is a bipartite graph, whose two sets of vertices consist of the classes of α and the classes of β, and the edges are the elements of S, each element connects its α-class with its β-class.  For each triple (a, b, r) of integers there is a finite set S with two partitions α and β on it such that each α-class has a elements, each β-class has b elements, an α-class can meet a β-class in at most one element, and in the incidence graph of α and β each simple cycle is longer than 2r. Corollary 3.3. Let α, β and S be as in Proposition 3.2. Suppose that the vertices of a graph are in S and the endpoints of each edge are either in the same α-class or in the same β-class. Then every simple cycle of length less than 2r is either entirely in a single α-class or entirely in a single β-class.
Proof. Follows immediately from Definition 3.1 Definition 3.4. Let A be a colored graph and Z a subset of the colors used in A. Then A Z denotes the subgraph obtained from A by omitting all edges whose color does not belong to Z. Suppose that C is another colored graph and φ is an r-isomorphism between A Z and C Z . We build another graph called the enhancement of C with A along φ, denoted by A φ ⇒ C. We start from C and add new edges to it. Namely, for each edge a → b of A whose color does not belong to Z we add a new edge φ(a) → φ(b) of the same color, provided that φ is defined at the endpoints a and b.
Lemma 3.5. Let Γ be a finitely generated group,X be a finite symmetric generating system of Γ and Z ⊆ X. Let A be a 2r-approximation of Cay(Γ, X) such that every vertex has degree at most d. Then there is a bound R enh (r, d) with the following property. Suppose that C is another colored graph wit color set Y such that X ∩ Y = Z and φ is an R enh (r, d)-isomorphism between A Z and C Z .
Proof. Clear from the definition. Lemma 3.6. Let G and H be finitely generated sofic groups, Γ ≤ G a finitely generated amenable subgroup and φ : Γ ֒→ H an injective homomorphism. Then the amalgamated product G * φ H is also sofic.
Proof. For simplicity we identify both G and H with their canonical image in G * φ H, this will also identify both Γ and φ(Γ) with G ∩ H. Let X ⊆ G, Y ⊆ H and Z ⊆ Γ be finite symmetric generating systems such that Z = X ∩ Y . Let d be the maximum of |X| and |Y |. The following is clear from the definition: Claim 3.7. For each r there is a constant R prod (r) with the following property. Suppose we find a colored graph C with colors from X ∪ Y such that C X is an R prod (r)-approximation of Cay(G, X), C Y is an R prod (r)-approximation of Cay(H, Y ), and every simple cycle of length at most 2r + 1 has either only X-colored edges or only Y -colored edges. Then C is an r-approximation of the Cay G * φ H, X ∪ Y .
We choose any integer r > 0 and set Let us start with r 4 -approximations A resp. B of the Cayley graphs Cay(G, X) resp. Cay(H, Y ). Then A Z and B Z be are both r 4 -approximations of Cay(Γ, Z). We choose a Følner sequence in Cay(Γ, Z). Then Proposition 2.8 provides a colored graph D (denoted there by α · T ) such that both A Z and B Z are r 3 -isomorphic to integer multiples a · D and b · D (with appropriate integers a, b > 0).
Applying Proposition 3.2 for the triple (a, b, r + 1) we obtain a finite set S and partitions α and β of S. Let S 0 denote the graph with no edge of vertex set S. Then the graph C 0 = S 0 × D is the union of |S| disjoint copies of D, namely the subgraphs {s} × D for s ∈ S. Now for each α-class σ ⊂ S the subgraph spanned by the subset σ × D is r 3 -isomorphic to A Z , so we can enhance it with A along this r 3 -isomorphism. Repeating this enhancement for all α-classes we obtain a new graph C 1 on the same vertex set S × D. According to Lemma 3.5, this C 1 is r 2 -isomorphic to |S|/a · A, and C 1 Z is still isomorphic to |S| · D.
Next for each β-class ρ ⊂ S we consider the subgraph C ρ 1 ⊂ C 1 spanned by the subset ρ× D. Then C ρ 1 Z is r 3 -isomorphic to B Z , so we can enhance it with B. Repeating this enhancement for all β-classes we obtain a new graph C 2 on the same vertex set S × D. According to Lemma 3.5, this C 2 X is r 2 -isomorphic to |S|/a · A and C 2 Y is r 2 -isomorphic to |S|/b · B.
We know from Lemma 2.2 that C 2 X is an r 1 -approximation of Cay(G, X) and C 2 Y is an r 1 -approximation of Cay(G, Y ). Let γ be a simple cycle of length at most 2r + 1 of C 2 . Then by Lemma 3.3, γ lies entirely in a single α-class or it lies entirely in a single β class. According to Claim 3.7, C 2 is an r-approximation of Cay G * φ H, X ∪ Y . Hence the lemma follows.
Proof. (of the main theorem) Let G, H be arbitrary sofic groups, Γ be an amenable subgroup of G and let φ : Γ → H be an injective homomorphism. Then the amalgamated product G ⋆ φ H is the direct limit of amalgamated products {G α ⋆ φ H α } over amenable subgroups Γ α , where G α , H α and Γ α are all finitely generated groups. Since the direct limits of sofic groups are sofic, G ⋆ φ H is a sofic group.

Hyperfiniteness
The notion of hyperfinite graph classes was introduced in [3]. A sofic approximation {G n } ∞ n=1 is hyperfinite if for any ǫ > 0 there exists K ǫ > 0 such that for any n ≥ 1 one can erase ǫ|E(G n )| edges of G n to obtain a graph G ′ n having components of size not greater than K ǫ . where L(A) is the set of edges having one endpoint in A and an other endpoint in A c . It is well-known that i(Cay(Γ, S)) = c > 0. Now let {G n } ∞ n=1 be a sofic approximation of Cay(Γ, S) and suppose that {G n } ∞ n=1 is hyperfinite. Let {G ′ n } ∞ n=1 be a graph sequence having components of size at most K such that each G ′ n is obtained by erasing c 10 |E(G n )| edges from G n . Observe that at least half of the vertices of G n are in such a component A of G ′ n such that Notice that the K-neighborhood of such vertices can not be isomorphic to the K-neighborhood of the unit element in Cay(Γ, S). This is in contradiction with the assumption that {G n } ∞ n=1 is a sofic approximation.

The proof of Theorem 2
Let ψ, φ : Γ → Σ be conjugate faithful embeddings of the group Γ. Let [ψ], [φ] : Γ → ∞ n=1 S n be representatives of the maps ψ resp. φ. This gives us for each n ≥ 1 the S-colored graphs G n and H n on the set {1, 2, . . . , n}. By the conjugacy assumption, for any r ≥ 1 there exists subsequences {G n k } ∞ k=1 , {H n k } ∞ k=1 such that • For any k ≥ 1, G n k and H n k are r-isomorphic to each other.
• Both G n k and H n k are r-approximations of Cay(Γ, S).
n=1 be a sofic approximation of Cay(Γ, S) such that |V (G n )| = n and Γ is nonamenable. By Proposition 4.1, we can suppose that there exists ǫ > 0 and a function r : N → N tending to infinity such that; if we erase ǫ|E(G n )| edges from G n , then at least one of the remaining components has size at least r(n). Now consider the sofic approximation {H n } ∞ n=1 , where H n consists of [ n r(n) ] disjoint copies of G rn and n − [ n r(n) ] single vertices. Let ψ resp. φ : Γ → Σ be the embeddings associated to the sofic approximations {G n } ∞ n=1 resp. {H n } ∞ n=1 . If ψ and φ are conjugate then by the previous arguments there exists n k > 0 such that G n k and H n k are [ 1 2ǫ ]-isomorphic to each other. Therefore, one can erase ǫ|E(G n k )| edges from G n k to obtain a graph having components of size at most r(n k ), leading to a contradiction. Thus there exist φ, ψ : Γ → Σ nonconjugate faithful embeddings provided that Γ in nonamenable. Now let Γ be amenable and ψ, φ : Γ → Σ be embeddings of Γ. Let {G n } ∞ n=1 and {H n } ∞ n=1 be the associated S-graphs (for representatives [ψ] and [φ]). For n ≥ 1, let h(n) be the largest integer for which G n and H n are both h(n)approximations of Cay(Γ, S). Using the definition of the ultraproduct it is easy to see that for any t ≥ 1 By Proposition 2.8, there exists a sofic approximation {J n } ∞ n=1 such that • |V (J n )| = n.
• G n and J n resp. H n and J n are 2r-isomorphic, whenever h(n) is large enough.
Therefore G n and H n are r-isomorphic if h(n) is large enough. Now we construct elements g ∈ ω S n such that gψg −1 = φ. For each n ≥ 1 let l(n) be the largest integer such that G n and H n are l(n)-isomorphic. Let g n ∈ S n be a bijection such that g n (G ′ n ) = H ′ n , where G ′ n resp H ′ n are isomorphic subgraphs obtained by erasing 1 ln |E(G n )| resp. 1 ln |E(H n )| edges from G n resp. H n . Then obviously gψg −1 = φ.