Growth of finite structures

[JOSEPH SHIPMAN]

Define GG(n) to be the cardinality of the largest finite group which has a presentation by generators and relations with the total length of the relations = n.

It seems to me that this function might have BusyBeaverish growth, but are any results known about it? It has the advantage of being much less arbitrary than the BB function, but the disadvantage of not getting big as fast (BB(6) and BB(7) are already known to be enormous).

In particular, is GG(n) known to eventually dominate any exponential function? (You can get exponential growth with cyclic groups; but presenting alternating or symmetric groups seems to require presentations of size quadratic in n while only giving factorial type growth, so cyclic is better).

What other kinds of finitely describable objects exhibit explosive growth while being less arbitrary than the BB function?

— JS