See [Grushko decomposition theorem][1].

In the cocompact case it is not true, since the group is not a free group and cannot be splitted into a non-trivial free product.

For convex-cocompact but not cocompact I know of particular examples with afirmative answer. Is it always the case?


  [1]: https://en.wikipedia.org/wiki/Grushko_theorem#Grushko_decomposition_theorem