Finite number of double cosets in a free product with amalgamation

Abstract:

If H is a finitely generated subgroup of a free group G such that every conjugate of H contains a cyclically reduced word then $(G:H) < \infty$. This generalizes a well-known result of Karrass and Solitar. If H is a finitely generated subgroup of the free product with amalgamation $G = A\;{ \ast _U}B$ such that every conjugate of H meets A and B trivially and contains a cyclically reduced word then G has only finitely many (H, U)-double cosets. Both theorems are proved by defining an action of G on a tree such that H acts freely.