Systolic growth of linear groups
  We prove that the residual girth of any finitely generated linear group is at most exponential. This means that the smallest finite quotient in which the $n$-ball injects has at most exponential size. If the group is also not virtually nilpotent, it follows that the residual girth is precisely exponential. 
Khalid Bou-Rabee and Yves Cornulier
math.GR