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)
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