The Ping-Pong Lemma: old and new

The classical ping-pong lemma, first discovered by Felix Klein, is a fundamental tool in group theory to detect free subgroups in a given group. For instance, it is central in the celebrated Tits' Alternative. Generalizations to amalgamated products and HNN extensions were formulated by Fenchel and Nielsen in unpublished notes that were circulating in the 50s, and were used in later important developments of geometric and combinatorial group theory, such as Stallings's theory or Maskit combination theorems for Kleinian groups. Motivated by a long-term project which aims at a classification of group actions on the circle, we formulate a generalization of the ping-pong lemma for fundamental groups of graph of groups. Joint work with J. Alonso, S. Alvarez, D. Malicet, C. Menino.