Degrees of growth of finitely generated groups, and the theory of invariant means.

*(English. Russian original)*Zbl 0583.20023
Math. USSR, Izv. 25, 259-300 (1985); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 48, No. 5, 939-985 (1984).

Let G be a finitely generated group, with a system of generators \(A=\{a_ 1,\ldots,a_ m\}\). By the growth function of G relative to the system A is meant the function \(\gamma\) (n), with argument a natural number n, such that \(\gamma\) (n) is equal to the number of distinct elements of G that are representable as a product of at most n factors of the form \(a_ i^{\delta}\), \(\delta =\pm 1\). On the set of growth functions may be defined a preorder \(\preccurlyeq\) and an equivalence relation \(\sim\) as follows: \(\gamma_ 1(n)\preccurlyeq \gamma_ 2(n)\) if there exists a natural number c such that \(\gamma_ 1(n)\leq \gamma_ 2(cn)\), \(n=1,2,\ldots\); and \(\gamma_ 1(n)\sim \gamma_ 2(n)\) if \(\gamma_ 1(n)\preccurlyeq \gamma_ 2(n)\) and \(\gamma_ 2(n)\preccurlyeq \gamma_ 1(n)\). For a given finitely generated group G the equivalence class of the growth function is independent of the system of generators and is called the degree of growth of the group G. J. Milnor [Am. Math. Mon. 75, 685-686 (1968)] posed the following question: is the growth function \(\gamma\) (n) of every finitely generated group equivalent either to a power function \(n^ d\) or to the exponential function \(2^ n?\)

The main result of the paper answers this question negatively: there exists a continuum of non-isomorphic periodic groups with two generators whose growth functions are not equivalent to any power function or to the exponential function (Theorem A). Concerning the degrees of growth of finitely generated groups, the following information is obtained, too. (1) There exists a finitely generated group G whose growth function \(\gamma\) (n) has the bounds \(2^{\sqrt{n}}\preccurlyeq \gamma (n)\preccurlyeq 2^{n^{\log_{32}31}}\) and satisfies the condition \(\gamma (n)\sim \gamma^ 2(n)\). (2) Suppose the function \(\rho\) (n) is such that \(\rho (n)=o(2^{\epsilon n})\) for arbitrary \(\epsilon >0\). Then there exists a finitely generated group G whose growth function \(\gamma\) (n) grows not more slowly than \(\rho\) (n), while \(\gamma\) (n) is not equivalent to \(2^ n\). (3) There exists a continuum of degrees of growth of groups. (Theorem B). Theorem A implies also that there exists a continuum of non-elementary amenable periodic groups with two generators which solves a question of M. Day [Ill. J. Math. 1, 509-544 (1957; Zbl 0078.294)]. Simultaneously, the results of the paper solve the Problems 4.5.a) and 8.6 from ”The Kourovka Notebook” [Novosibirsk (1982; Zbl 0509.20001), see also the English translation Transl., II. Ser., Am. Math. Soc. 121 (1983; Zbl 0512.20001)]. A number of other results are obtained on residually finite finitely generated infinite 2-groups.

The main result of the paper answers this question negatively: there exists a continuum of non-isomorphic periodic groups with two generators whose growth functions are not equivalent to any power function or to the exponential function (Theorem A). Concerning the degrees of growth of finitely generated groups, the following information is obtained, too. (1) There exists a finitely generated group G whose growth function \(\gamma\) (n) has the bounds \(2^{\sqrt{n}}\preccurlyeq \gamma (n)\preccurlyeq 2^{n^{\log_{32}31}}\) and satisfies the condition \(\gamma (n)\sim \gamma^ 2(n)\). (2) Suppose the function \(\rho\) (n) is such that \(\rho (n)=o(2^{\epsilon n})\) for arbitrary \(\epsilon >0\). Then there exists a finitely generated group G whose growth function \(\gamma\) (n) grows not more slowly than \(\rho\) (n), while \(\gamma\) (n) is not equivalent to \(2^ n\). (3) There exists a continuum of degrees of growth of groups. (Theorem B). Theorem A implies also that there exists a continuum of non-elementary amenable periodic groups with two generators which solves a question of M. Day [Ill. J. Math. 1, 509-544 (1957; Zbl 0078.294)]. Simultaneously, the results of the paper solve the Problems 4.5.a) and 8.6 from ”The Kourovka Notebook” [Novosibirsk (1982; Zbl 0509.20001), see also the English translation Transl., II. Ser., Am. Math. Soc. 121 (1983; Zbl 0512.20001)]. A number of other results are obtained on residually finite finitely generated infinite 2-groups.

Reviewer: Yu.I.Merzlyakov

##### MSC:

20F05 | Generators, relations, and presentations of groups |

20F50 | Periodic groups; locally finite groups |

20E26 | Residual properties and generalizations; residually finite groups |

43A07 | Means on groups, semigroups, etc.; amenable groups |