12.56 (1992)

Let $\langle X \mid R \rangle$ be a finite presentation (the words in $R$ are assumed cyclically reduced). Define the length of the presentation to be the sum of the number of generators and the lengths of the relators. Let $F(n)$ be the number of (isomorphism types of) groups that have a presentation of length at most $n$. What can one say about the function $F(n)$? It can be shown that $F(n)$ is not recursive (D. Segal) and it is at least exponential (L. Pyber).