Non-finitely generated relatively hyperbolic groups

Question

As we know that free product $G_1\ast G_2$ is relatively hyperbolic with respect to $\{G_1,G_2\}$. So for non-finitely generated relatively hyperbolic groups we can take this example. But i am looking for other than this example of relatively hyperbolic groups which is not finitely generated.

Thanks in advance for any help.

Answer 1

If you ignore any finitely generated conditions then the group of all permutations on a countable set (any cardinal actually), $S_\mathbb{N}$, is a hyperbolic group. This means that it is relatively hyperbolic with respect to the identity subgroup. This is proved by George Bergman in Generating infinite symmetric groups where it is shown that any generating set gives a bounded diameter Cayley graph.

Answer 2

Non-finitely generated groups which are relatively hyperbolic are studied in great details in a paper by V. Gerasimov and L. Potyagailo, available here : https://arxiv.org/abs/1008.3470