en.wikipedia.org/wiki/Sofic_group
1 correction found
every two elements of the group have distance 1
This characterization is misstated. In the ultraproduct formulation, the distance-1 condition applies to distinct elements (or, equivalently, to every nonidentity element versus the identity), not to every pair of elements.
Full reasoning
As written, this cannot be correct: in any metric space, an element has distance 0 from itself, not 1. So the phrase "every two elements" is false when the two elements are the same.
Standard formulations of the ultraproduct characterization of sofic groups use one of these equivalent statements instead:
- every two distinct elements in the embedded image are at distance 1 from each other; or
- every nonidentity element is at distance 1 from the identity.
A survey by Pestov states the characterization as embedding into an ultraproduct of finite symmetric groups "in such a way that every two distinct elements in the image are at a distance 1 from each other." A 2025 Research in the Mathematical Sciences paper likewise gives the usual finitary definition: for nonidentity elements, the approximating permutations must stay almost distance 1 from the identity, i.e. (d(1,\varphi(g))>1-\epsilon) for all (g\neq e).
So the article is missing the crucial qualifier distinct (or equivalently nontrivial/nonidentity).
2 sources
- An Introduction to Hyperlinear and Sofic Groups
Theorem 5.2: a group is sofic iff it embeds into an ultraproduct of symmetric groups of finite rank "in such a way that every two distinct elements in the image are at a distance 1 from each other."
- Soficity for group actions on sets and applications
Definition 2.1(4): G is sofic if for all finite F and ε>0 there is φ:G→Sym(A) with d(1, φ(g)) > 1 − ε for all g in F\setminus\{e\}.