A group satisfies the "Tits alternative" if every subgroup is either virtually solvable or contains a nonabelian free group. This is named after J. Tits who proved that all finitely generated linear groups enjoy this property. The Tits alternative was established for braid groups by Ivanov and McCarthy, but now also follows from linearity (due to Bigelow-Krammer). I'll discuss joint work with D. Margalit, in which we prove a strong version of the Tits alternative for the pure braid groups: every two elements of the pure braid group either commute or generate a free group. The proof uses 3-manifold topology and actions on trees.