Polynomial and Exponential Growth
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 virtually nilpotent, 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.
Read more about this topic: Growth Rate (group Theory)
Famous quotes containing the word growth:
“Cities force growth and make men talkative and entertaining, but they make them artificial. What possesses interest for us is the natural of each, his constitutional excellence. This is forever a surprise, engaging and lovely; we cannot be satiated with knowing it, and about it; and it is this which the conversation with Nature cherishes and guards.”
—Ralph Waldo Emerson (18031882)