[math-fun] matrix group with bounded eigenvalues

Given a finite set of complex matrices generating an infinite group, I need to establish that every element of the group has all eigenvalues within the unit circle. Such problems must have been well investigated --- perhaps (shudder) by math physicists --- but I have no idea where to start looking. [It's actually a semigroup of integer matrices with constraints on allowed products, and circle radius exponential in the number of product factors --- but those details are almost certainly unimportant.] WFL