Ping-pong for basis-conjugating HNN-extension of free group

Abstract

We isolate a tractable class of HNN-extensions of a free group, namely, multiple HNN-extensions by basis-conjugating embeddings. For this class, we construct a normal form and establish a practical version of the ping-pong lemma that provides verifiable sufficient conditions for a set of elements to generate a free subgroup. We then apply these results to the pure braid group $P_{n+1}$, exploiting its well-known decomposition as a semidirect product of free groups. Our approach yields new families of free subgroups within the first two factors $F_n \rtimes F_{n-1}$ of this decomposition.