mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a sofic group is a

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...

whose

Cayley graph In mathematics, a Cayley graph, also known as a Cayley color graph, Cayley diagram, group diagram, or color group is a graph that encodes the abstract structure of a group. Its definition is suggested by Cayley's theorem (named after Arthur Cayle ...

is an initially subamenable graph, or equivalently a

subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...

of an

ultraproduct The ultraproduct is a mathematical construction that appears mainly in abstract algebra and mathematical logic, in particular in model theory and set theory. An ultraproduct is a quotient of the direct product of a family of structures. All factors ...

of finite-rank

symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group \m ...

s such that every two elements of the group have distance 1.Ceccherini-Silberstein & Coornaert (2010) p. 276 They were introduced by as a common generalization of amenable and

residually finite group {{unsourced, date=September 2022 In the mathematical field of group theory, a group ''G'' is residually finite or finitely approximable if for every element ''g'' that is not the identity in ''G'' there is a homomorphism ''h'' from ''G'' to a finit ...

s. The name "sofic", from the

Hebrew Hebrew (; ; ) is a Northwest Semitic language of the Afroasiatic language family. Historically, it is one of the spoken languages of the Israelites and their longest-surviving descendants, the Jews and Samaritans. It was largely preserved ...

word meaning "finite", was later applied by , following Weiss's earlier use of the same word to indicate a generalization of finiteness in

sofic subshift In symbolic dynamics and related branches of mathematics, a shift space or subshift is a set of infinite words that represent the evolution of a discrete system. In fact, shift spaces and '' symbolic dynamical systems'' are often considered synonyms ...

s. The class of sofic groups is

closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...

under the operations of taking subgroups,

extensions Extension, extend or extended may refer to: Mathematics Logic or set theory * Axiom of extensionality * Extensible cardinal * Extension (model theory) * Extension (predicate logic), the set of tuples of values that satisfy the predicate * Ex ...

by amenable groups, and

free product In mathematics, specifically group theory, the free product is an operation that takes two groups ''G'' and ''H'' and constructs a new The result contains both ''G'' and ''H'' as subgroups, is generated by the elements of these subgroups, and i ...

s. A

finitely generated group In algebra, a finitely generated group is a group ''G'' that has some finite generating set ''S'' so that every element of ''G'' can be written as the combination (under the group operation) of finitely many elements of ''S'' and of inverses of s ...

is sofic if it is the

limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...

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.. As Gromov proved, Sofic groups are surjunctive. That is, they obey a form of the

Garden of Eden theorem In a cellular automaton, a Garden of Eden is a configuration that has no predecessor. It can be the initial configuration of the automaton but cannot arise in any other way. John Tukey named these configurations after the Garden of Eden in Abr ...

for

cellular automata A cellular automaton (pl. cellular automata, abbrev. CA) is a discrete model of computation studied in automata theory. Cellular automata are also called cellular spaces, tessellation automata, homogeneous structures, cellular structures, tessel ...

defined over the group (

dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...

s whose states are mappings from the group to a

finite set In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, :\ is a finite set with five elements. Th ...

and whose state transitions are translation-invariant and continuous) stating that every injective automaton is surjective and therefore also reversible.Ceccherini-Silberstein & Coornaert (2010) p. 56

Notes

References

*. *. *. *{{citation , last = Weiss , first = Benjamin , authorlink = Benjamin Weiss , issue = 3 , journal = Sankhyā , mr = 1803462 , zbl=1148.37302 , pages = 350–359 , series = Series A , title = Sofic groups and dynamical systems , url = http://sankhya.isical.ac.in/search/62a3/eh06fnl.pdf , volume = 62 , year = 2000. Properties of groups