This question is in reference to this other question,

Can someone point out references (or explain!) which give techniques of being able to prove for any Cayley graph this property of having a girth logarithmic in size of the group?

The only example like this that I can see is this argument in theorem 3.4 in this paper, http://math1.math.huji.ac.il/~alexlub/PAPERS/ramanujan%20graphs/ramanujanGraphs.pdf but this looks like an extremely specialized calculation and its not clear to me if anything here can be done in other situations. I would like to to know if there are any generic insights known!

I am guessing that there will be a difference in the techniques depending on which of the 3 scenarios in the linked MO question is one trying to address. Given a non-Abelian group proving that (1) there exists a set of generators with this property will possibly entail a totally different proof than proving that (2) any arbitrarily picked large enough symmetric generating set or (3) a symmetric generating set picked uniformly at random has this logarithmic girth property.

Also if someone can point out methods about finding one such symmetric generating set with this property in cases where any of the three scenarios is true!

This question is in reference to this other question,

Can someone point out references (or explain!) which give techniques of being able to prove for any Cayley graph this property of having a girth logarithmic in size of the group?

The only example like this that I can see is this argument in theorem 3.4 in this paper, http://math1.math.huji.ac.il/~alexlub/PAPERS/ramanujan%20graphs/ramanujanGraphs.pdf but this looks like an extremely specialized calculation and its not clear to me if anything here can be done in other situations. I would like to to know if there are any generic insights known!

I am guessing that there will be a difference in the techniques depending on which of the 3 scenarios in the linked MO question is one trying to address. Given a non-Abelian group proving that (1) there exists a set of generators with this property will possibly entail a totally different proof than proving that (2) any arbitrarily picked large enough symmetric generating set or (3) a symmetric generating set picked uniformly at random has this logarithmic girth property.

Also if someone can point out methods about finding one such symmetric generating set with this property in cases where any of the three scenarios is true!