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Sofic group

Unsolved problem in mathematics

Is every group sofic?

Two balls of radius 2 in the Cayley graph of the dihedral group $D_{4}$ . Both balls contain 7 of the 8 group elements and are isomorphic as edge-colored graphs. Furthermore, for *any* radius $r$ and $\varepsilon >0$ , we can use the approximating graph $G_{0}$ as the full Cayley graph itself, where the set $W$ of all vertices satisfies $|W|/|V|=8/8=1>1-\varepsilon$ . Since all 8 vertices have isomorphic $r$ -balls (by the vertex-transitivity of Cayley graphs), $D_{4}$ is sofic.

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 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. 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]

Notes

1 2 Ceccherini-Silberstein & Coornaert (2010) p. 276
↑ Cornulier (2011).
↑ Ceccherini-Silberstein & Coornaert (2010) p. 56

References

Ceccherini-Silberstein, Tullio; Coornaert, Michel (2010), Cellular Automata and Groups, Springer Monographs in Mathematics, Springer-Verlag, doi:10.1007/978-3-642-14034-1, ISBN 978-3-642-14033-4, MR 2683112, Zbl 1218.37004.
Cornulier, Yves (2011), "A sofic group away from amenable groups", Mathematische Annalen, 350 (2): 269–275, arXiv:0906.3374, doi:10.1007/s00208-010-0557-8, MR 2794910, S2CID 12966793, Zbl 1247.20039.
Gromov, M. (1999), "Endomorphisms of symbolic algebraic varieties", Journal of the European Mathematical Society, 1 (2): 109–197, doi:10.1007/PL00011162, MR 1694588, Zbl 0998.14001.
Weiss, Benjamin (2000), "Sofic groups and dynamical systems" (PDF), Sankhyā, Series A, 62 (3): 350–359, MR 1803462, Zbl 1148.37302.