For more recent work than that mentioned by Alain, check out this modern classic by Breuillard, Green, Guralnick, Tao

Let's try again. For $SL(2),$ there is an argument due to Bourgain-Gamburd, which can be found in these notes of Emmanuel Breuillard. (corollary 0.2). Other gith estimates are shown in the well-known paper of Gamburd, Hoory, Shahshahani, Shalev, Virag

MR2532876
Gamburd, A.(1-UCSC); Hoory, S.(IL-IBM); Shahshahani, M.(IR-TPM-SM); Shalev, A.(IL-HEBR-   IM); Virág, B.(3-TRNT-MS)
On the girth of random Cayley graphs. (English summary) 
Random Structures Algorithms 35 (2009), no. 1, 100–117. )

Both the results and the techniques (to give a girth estimate) are of interest to the OP, and to others, I assume.