On the growth functions of finitely generated groups

For a group $G$ generated by some finite set $S$, the growth function $f_{G,S}(n)$ is defined as the number of distinct elements of the group that can be written as a product $a_1 a_2 \ldots a_n$, where $a_i\in S\cup S^{-1}\cup \{1\}$. In other words, this function is equal to the number of vertices in the ball of radius $n$ in the Cayley graph $Cay(G,S)$. Over the past 50 years, a number of results have been obtained in this area, including the famous theorem by M. Gromov's on the structure of groups having polynomial growth and examples of groups having intermediate growth constructed by R.I.Grigorchuk. I will review these and some other interesting results about growth functions and also describe the connection of growth functions of Coxeter groups to finite automata, Perron-Frobenius theory and Pisot and Salem algebraic numbers.