Unsolved problem in mathematics :
Is every group 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
^ a b 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 .