[1] They were introduced by Gromov (1999) as a common generalization of amenable and residually finite groups.
The name "sofic", from the Hebrew word סופי meaning "finite", was later applied by Weiss (2000), 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.
[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.