In group theory, the growth rate of a group with respect to a symmetric generating set describes the size of balls in the group. Every element in the group can be written as a product of generators, and the growth rate counts the number of elements that can be written as a product of length n. Group theory is that branch of mathematics concerned with the study of groups. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In mathematics, a subset S of a algebraic structure G is a generating set of G (or G is generated by S) if the smallest subset of G that includes S and is closed under the algebraic operations on G is G itself. ...

Definition

Suppose G is a finitely generated group; and T is a finite symmetric set of generators (symmetric means that if then ). Any element can be expressed as a word in the T-alphabet In abstract algebra, a generating set of a group is a subset S such that every element of G can be expressed as the product of finitely many elements of S and their inverses. ... In theoretical computer science, automata theory is the study of abstract machines and problems they are able to solve. ...

Let us consider the subset of all elements of G which can be presented by such a word of length ≤n

This set is just the closed ball of radius n in the word metric d on G with respect to the generating set T: The solid interior of a sphere or circle; in mathematics, latter terms refer specifically to the (n-1)-dimensional surface of an n-dimensional solid ball. ... In mathematics, the word metric is a metric defined on a group, depending on a set of generators for the group. ...

More geometrically, B_n(G,T) is the set of vertices in the Cayley graph with respect to T which are within distance n of the identity. The Cayley graph of the free group on two generators a and b In mathematics, a Cayley graph, named after Arthur Cayley, is a graph that encodes the structure of a group. ...

Given two nondecreasing positive functions a and b one can say that they are equivalent () if there is a constant C such that

for example if p,q > 1.

Then the growth rate of the group G can be defined as the corresponding equivalence class of the function In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x ∈ X | x ~ a } The notion of equivalence classes is useful for constructing sets out...

where | B_n(G,T) | denotes the number of elements in the set B_n(G,T). Although the function depends on the set of generators T its rate of growth does not (see below) and therefore the rate of growth gives an invariant of a group.

The word metric d and therefore sets B_n(G,T) depend on the generating set T. However, any two such metrics are bilipschitz equivalent in the following sense: for finite symmetric generating sets E, F, there is a positive constant C such that In mathematics, a function f : D → R defined on a set D of real numbers with real values is called Lipschitz continuous (or is said to satisfy a Lipschitz condition) if there exists a constant K ≥ 0 such that for all in D. The smallest such K is called the... In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x ∈ X | x ~ a } The notion of equivalence classes is useful for constructing sets out...

As an immediate corollary of this inequality we get that the growth rate does not depend on the choice of generating set.

Polynomial and exponential growth

If for some we say that G has a polynomial growth rate. The infimum k₀ of such k's is called the order of polynomial growth. According to Gromov's theorem, a group of polynomial growth is almost nilpotent, i.e. it has a nilpotent subgroup of finite index. In particular, the order of polynomial growth k₀ has to be a natural number and in fact . In mathematics, Gromovs theorem on groups of polynomial growth, named for Mikhail Gromov, characterizes finitely generated groups of polynomial growth, as those groups which have nilpotent subgroups of finite index. ... In group theory, a nilpotent group is a group having a special property that makes it almost abelian, through repeated application of the commutator operation, [x,y] = x-1y-1xy. ... In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group... In mathematics, if G is a group, H a subgroup of G, and g an element of G, then gH = { gh : h an element of H } is a left coset of H in G, and Hg = { hg : h an element of H } is a right coset of H in G... Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is...

If for some a > 1 we say that G has an exponential growth rate. Every finitely generated G has at most exponential growth, i.e. for some b > 1 we have . In mathematics, a quantity that grows exponentially is one whose growth rate is always proportional to its current size. ... In abstract algebra, a generating set of a group is a subset S such that every element of G can be expressed as the product of finitely many elements of S and their inverses. ...

If grows more slowly than any exponential function, G has a subexponential growth rate. Any such group is amenable. In mathematics, an amenable group is a topological group G carrying a kind of averaging operation, that is invariant under translations by group elements. ...

Examples

• A free group with a finite rank k > 1 has an exponential growth rate.

• If M is a closed negatively curved Riemannian manifold then its fundamental group π₁(M) has exponential growth rate. Milnor proved this using the fact that the word metric on π₁(M) is quasi-isometric to the universal cover of M.

• Z² has a polynomial growth rate of order 2.

• The discrete Heisenberg group H₃ has a polynomial growth rate of order 4. This fact is a special case of the general theorem of Bass that is discussed in the article on Gromov's theorem.

• The lamplighter group has an exponential growth. This is a rare example of a solvable group with exponential growth.

• The existence of groups with intermediate growth, i.e. subexponential but not polynomial was open for many years. It was asked by Milnor in 1968 and was finally answered in the positive by Grigorchuk in 1984. There are still open questions in this area and a complete picture of which orders of growth are possible and which are not is missing.

The Cayley graph of the free group on two generators a and b In mathematics, a group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many... In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ... In mathematics, the fundamental group is one of the basic concepts of algebraic topology. ... John Willard Milnor (b. ... In mathematics, the word metric is a metric defined on a group, depending on a set of generators for the group. ... This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesnt cover the terminology of differential topology. ... In mathematics, specifically topology, a covering map is a continuous surjective map p : C → X, with C and X being topological spaces, which has the following property: to every x in X there exists an open neighborhood U such that p -1(U) is a union of mutually disjoint open... In mathematics, the Heisenberg group is a group of 3×3 upper triangular matrices of the form Elements a,b,c can be taken from some (arbitrary) commutative ring. ... In mathematics, Gromovs theorem on groups of polynomial growth, named for Mikhail Gromov, characterizes finitely generated groups of polynomial growth, as those groups which have nilpotent subgroups of finite index. ... In mathematics, the lamplighter group L of group theory is the wreath product Z/2Z ≀ Z. The base group B of L is :, and so L/B is isomorphic to Z. Growth rate (group theory) Categories: | ... John Willard Milnor (b. ...

See also

Connections to isoperimetric inequalities In mathematics, the isoperimetric dimension of a manifold is a notion of dimension that tries to capture how the large scale behavior of the manifold resembles that of a Euclidean space (unlike the topological dimension or the Hausdorff dimension which compare different local behaviors against those of the Euclidean space). ...

References

• J. Milnor, A note on curvature and fundamental group, J. Differential Geometry 2 (1968), 1-7.
• R. I. Grigorchuk, Degrees of growth of finitely generated groups and the theory of invariant means., Izv. Akad. Nauk SSSR Ser. Mat. 48:5 (1984), 939-985 (Russian).
• Rostislav Grigorchuk and Igor Pak (2006). Groups of Intermediate Growth: an Introduction for Beginners. arXiv.