Given a non-Abelian group $G$ I want to choose a symmetric generating set $S \subset G$ such that $Cay(G,S)$ has girth logarithmic in the size of the set. I want to know,
- For which $G$ can the existence of such a $S$ be guaranteed?
- For which $G$ will any arbitrary choice of a large enough set $S$ guarantee this property?
- For which $G$ will a $S$ of large enough size picked uniformly at random be likely to ensure this property?
Some searching through literature tells me that $SL_2(\mathbb{F}_p)$ satisfies all the above properties. Any reference for proofs of this particular example would also be helpful. Though my question is more like looking for generic patterns.
I can't make this intuition precise but my gut feeling is that all these properties are true if the dimension of the smallest irrep of $G$ is of the order of some root of the size of the group. (like for $SL_2(\mathbb{F}_p)$ the smallest irrep is of dimension order of cube-root of the size of the group)
One might also be reminded of things like, http://www.combinatorics.org/ojs/index.php/eljc/article/view/v11i1r62/pdf http://www.eatcs.org/beatcs/index.php/beatcs/article/view/209