Editing Growth rate (group theory) (section)

==Polynomial and exponential growth==<!-- This section is linked from [[Hyperbolic geometry]] -->

If

:<math>\#(n)\le C(n^k+1)</math>

for some <math>C,k<\infty</math> we say that ''G'' has a '''polynomial growth rate'''.
The infimum <math>k_0</math> of such ''k'''s is called the '''order of polynomial growth'''.
According to [[Gromov's theorem on groups of polynomial growth|Gromov's theorem]], a group of polynomial growth is a [[virtually]] [[nilpotent group]], i.e. it has a [[nilpotent group|nilpotent]] [[subgroup]] of finite [[Index of a subgroup|index]].  In particular, the order of polynomial growth <math>k_0</math> has to be a [[natural numbers|natural number]] and in fact <math>\#(n)\sim n^{k_0}</math>.

If <math>\#(n)\ge a^n</math> for some <math>a>1</math> we say that ''G'' has an '''[[exponential growth]] rate'''.
Every [[finitely generated group|finitely generated]] ''G'' has at most exponential growth, i.e. for some <math>b>1</math> we have <math>\#(n)\le b^n</math>.

If <math>\#(n)</math> grows [[Infra-exponential|more slowly than any exponential function]], ''G'' has a '''subexponential growth rate'''. Any such group is [[amenable group|amenable]].