Residual properties of the modular group and other free products

Using a probabilistic approach we establish new residual properties of the modular group PSL₂ (ℤ), and of more general free products. We prove that the modular group is residually in any infinite collection of finite simple groups not containing a Suzuki group Sz(q) or a 4-dimensional symplectic group PS_p4(q) with q a power of 2 or 3. This result is best possible, since the groups excluded are not quotients of the modular group. We also show that if S is a collection of classical groups of unbounded rank, then an arbitrary free product A * B of nontrivial finite groups, not both 2-groups, is residually S, and prove results about free products A * ℤ.