Ping-pong and exceptional vector bundles

We present a strategy for proving that full exceptional collections of vector bundles on projective n-space can be constructed from a standard collection of line bundles, reducing the question of constructibility to the problem of freeness of a certain finitely generated linear group. We use the ping-pong lemma of Fricke-Klein to solve this problem in low dimensions, thus providing a new proof of constructibility of exceptional collections in some cases. We expect a similar ping-pong argument to give constructibility on projective n-space and on some other Fano varieties of Picard rank one. This is joint work in progress with Hugh Thomas.