Local Embeddability and Sofic Groups

Abstract

In this chapter we study the notions of local embeddability and soficity for groups. Roughly speaking, a group is locally embeddable into a given class of groups provided that the multiplicative table of any finite subset of the group is the same as the multiplicative table of a subset of some group in the class (cf. Definition 7.1.3). In Sect. 7.1 we discuss several stability properties of local embeddability. Subgroups of locally embeddable groups are locally embeddable (Proposition 7.1.7). Moreover, as the name suggests, local embeddability is a local property, that is, a group is locally embeddable into a class of groups if and only if all its finitely generated subgroups are locally embeddable into the class (Proposition 7.1.8). When the class is closed under finite direct products, the class of groups which are locally embeddable in the class is closed under (possibly infinite) direct products (Proposition 7.1.10). We also show that a marked group which is a limit of groups belonging to a given class is locally embeddable into this class (Theorem 7.1.16). Conversely, we prove that if the given class is closed under taking subgroups and the marking group is free, then any marked group which is locally embeddable is a limit of marked groups which are in the class (Theorem 7.1.19). If $\mathcal {C}$ is a class of groups which is closed under finite direct products, then every group which is residually $\mathcal {C}$ is locally embeddable into $\mathcal {C}$ (Corollary 7.1.14). Conversely, under the hypothesis that $\mathcal {C}$ is closed under taking subgroups, every finitely presented group which is locally embeddable into $\mathcal {C}$ is residually  $\mathcal {C}$ (Corollary 7.1.21).

In Sect. 7.2 we present a characterization of local embeddability in terms of ultraproducts: a group is locally embeddable into $\mathcal {C}$ if and only if it can be embedded into an ultraproduct of a family of groups in $\mathcal {C}$ .

Section 7.3 is devoted to LEF and LEA-groups. A group is called LEF (resp. LEA) if it is locally embeddable into the class of finite (resp. amenable) groups. As the class of finite (resp. amenable) groups is closed under taking subgroups and taking finite direct products, all the results obtained in the previous section can be applied. This implies in particular that every locally residually finite (resp. locally residually amenable) group is LEF (resp. LEA). We give an example of a finitely generated amenable group which is LEF but not residually finite (Proposition 7.3.9). This group is not finitely presentable since every finitely presented LEF-group is residually finite.

The Hamming metric, which is a bi-invariant metric on the symmetric group of a finite set, is introduced in Sect. 7.4.

In Sect. 7.5 we define the class of sofic groups. These groups are the groups admitting local approximations by finite symmetric groups equipped with their Hamming metric. Subgroups and direct products of sofic groups are sofic. Every LEA-group is sofic (Corollary 7.5.11). In particular, residually amenable groups, and therefore amenable groups and residually finite groups, are sofic. A group is sofic if and only if it can be embedded into an ultraproduct of a family of finite symmetric groups equipped with their Hamming metrics (Theorem 7.6.6). In Sect. 7.7 we give a characterization of finitely generated sofic groups in terms of their Cayley graphs. More precisely, we show that a finitely generated group G with a finite symmetric generating subset S⊂G is sofic if and only if, for every integer r≥0 and every ε>0, there exists a finite S-labeled graph Q such that there is a proportion of at least 1−ε of vertices q∈Q such that the ball of radius r centered at q in Q is isomorphic, as a labeled graph, to a ball of radius r in the Cayley graph of G associated with S (Theorem 7.7.1). The last section of this chapter is devoted to the proof of the surjunctivity of sofic groups (Theorem 7.8.1).