ABSTRACT
Groups of polynomial growth were described by Gromov. A finitely generated group G has polynomial growth if and only if it is nilpotent-by-finite, that is, G has a nilpotent subgroup N of finite index. Moreover, in this case, N is a finitely generated group, and the exponents of growth of N and G are equal. Therefore, the growth of G may be computed using Bass’s formula for finitely generated nilpotent groups. This chapter presents an extension of Gromov’s theorem to the class of cancellative semigroups. It proves that a cancellative semigroup of polynomial growth has no free noncommutative subsemigroups and cancellative semigroup has a group of fractions.
