A finitely generated group G is said to have a uniform radius of independence R if, for any finite generating set S of G, there exists two elements of S-word length at most R that generate a free nonabelian subgroup; this is a measure of the ubiquity of free subgroups inside G. A group is said to satisfy the uniform Tits alternative if its finitely generated subgroups are either virtually solvable, or have a uniform radius of independence. This property was established for linear groups by E. Breuillard and T. Gelander, extending the work of J. Tits.

 

I will explain how an analogous property can be stated for semigroups (where "free subgroup" is replaced with "free subsemigroup", and "virtually solvable" is replaced with an appropriate notion of "virtually nilpotent"), and sketch a proof of it for the semigroup of endomorphisms of the projective line. In particular, I will describe a ping-pong argument where arithmetic height functions make a remarkable appearance.