How to construct a group whose "size" grows between polynomially and exponentially.

Question

Define the "size" of a group with respect to an integer $n$ to be the number of possible reduced words that we can create by combining $n$ elements of the group. For example:

1. an infinite cyclic group has size $\Theta(n)$
2. the free abelian group with $a>1$ generators has size $\Theta(n^a)$
3. the free group with $a>1$ generators has size $\Omega(a^n)$

I'm interested in the fact that free abelian groups grow polynomially in size, whereas free groups grow exponentially in size. Is there a way to quotient the free group to get a growth rate between polynomial and exponential?

Also, is there a more standard term for what I'm calling the size of a group?

Answer 1

For the sake of having an answer (CW): as Derek Holt says in the comments, what you call size is called the growth rate, and the groups you want are called groups of intermediate growth. Milnor asked in 1968 whether such groups existed, and Grigorchuk answered the question in 1984 by constructing the Grigorchuk group and proving that it has intermediate growth.

See also Gromov's theorem on groups of polynomial growth.