Homeomorphism types of the Bowditch boundaries of infinite-ended relatively hyperbolic groups

For n ≥ 2 n\geq 2 , let G 1 = A 1 ∗ ⋯ ∗ A n G_{1}=A_{1}\ast\dots\ast A_{n} and G 2 = B 1 ∗ ⋯ ∗ B n G_{2}=B_{1}\ast\dots\ast B_{n} where the A i A_{i} ’s and B i B_{i} ’s are non-elementary relatively hyperbolic groups. Suppose that, for 1 ≤ i ≤ n 1\leq i\leq n , the Bowditch boundary of A i A_{i} is homeomorphic to the Bowditch boundary of B i B_{i} . We show that the Bowditch boundary of G 1 G_{1} is homeomorphic to the Bowditch boundary of G 2 G_{2} . We generalize this result to graphs of relatively hyperbolic groups with finite edge groups. This extends Martin–Świątkowski’s work in the context of relatively hyperbolic groups.