Sofic group

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 distinct elements of the group have distance 1.^[1] They were introduced by Gromov as a common generalization of amenable and residually finite groups.^[2] The name "sofic", from Hebrew סופי 'finite', was later applied by Weiss,^[3] following his use of the word as a generalization of finiteness in sofic subshifts. A non-sofic group is a group that is not a sofic group.

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.^[4]

Gromov proved that 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.^[5]