Growth rate (group theory)

In the mathematical subject of geometric group theory, the growth rate of a group with respect to a symmetric generating set describes how fast a group grows. 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.

DefinitionEdit

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

 

Consider the subset of all elements of G that can be expressed 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:

 

More geometrically,   is the set of vertices in the Cayley graph with respect to T that are within distance n of the identity.

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

 

for example   if  .

Then the growth rate of the group G can be defined as the corresponding equivalence class of the function

 

where   denotes the number of elements in the set  . 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   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

 

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 growthEdit

If

 

for some   we say that G has a polynomial growth rate. The infimum   of such k's is called the order of polynomial growth. According to Gromov's theorem, a group of polynomial growth is a virtually nilpotent group, i.e. it has a nilpotent subgroup of finite index. In particular, the order of polynomial growth   has to be a natural number and in fact  .

If   for some   we say that G has an exponential growth rate. Every finitely generated G has at most exponential growth, i.e. for some   we have  .

If   grows more slowly than any exponential function, G has a subexponential growth rate. Any such group is amenable.

ExamplesEdit

See alsoEdit

ReferencesEdit

  • Milnor J. (1968). "A note on curvature and fundamental group". Journal of Differential Geometry. 2: 1–7. doi:10.4310/jdg/1214501132.
  • Grigorchuk R. I. (1984). "Degrees of growth of finitely generated groups and the theory of invariant means". Izv. Akad. Nauk SSSR Ser. Mat. (in Russian). 48 (5): 939–985.

Further readingEdit