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. They were introduced by Gromov as a common generalization of amenable and residually finite groups. The name "sofic", from Hebrew סופי 'finite', was later applied by Weiss, 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. Gromov proved that Sofic groups are surjunctive. 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.
Existence of non-sofic groups In 2026, artificial intelligence research company OpenAI announced a machine-checkable proof by construction of the existence of a non-sofic countable discrete group. The existence of such groups was previously an open question.
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