In mathematics, a sofic group is a group whose Cayley graph is an initially subamenable graph, or equivalently a subgroup of an ultraproduct of finite-rank symmetric groups such that every two elements of the group have distance 1.[1] They were introduced by as a common generalization of amenable and residually finite groups. The name "sofic", from the Hebrew word Hebrew: סופי meaning "finite", was later applied by, following Weiss's earlier use of the same word to indicate a generalization of finiteness in sofic subshifts.
The class of sofic groups is closed under the operations of taking subgroups, extensions by amenable groups, and free products. A finitely generated group is sofic if it is the limit of a sequence of sofic groups. The limit of a sequence of amenable groups (that is, an initially subamenable group) is necessarily sofic, but there exist sofic groups that are not initially subamenable groups.[2]
As Gromov proved, Sofic groups are surjunctive.[1] That is, they obey a form of the Garden of Eden theorem for cellular automata defined over the group (dynamical systems whose states are mappings from the group to a finite set and whose state transitions are translation-invariant and continuous) stating that every injective automaton is surjective and therefore also reversible.[3]