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.][1] (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. 


  [1]: http://www.math.utah.edu/pcmi12/lecture_notes/breuillard4.pdf