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

Abstract For n ≥ 2 n 2, let G 1 = A 1 ∗ ⋯ ∗ A n G₁=A₁ A₍ and G 2 = B 1 ∗ ⋯ ∗ B n G₂=B₁ B₍ where the A i A₈ ’s and B i B₈ ’s are non-elementary relatively hyperbolic groups. Suppose that, for 1 ≤ i ≤ n 1 i n, the Bowditch boundary of A i A₈ is homeomorphic to the Bowditch boundary of B i B₈. We show that the Bowditch boundary of G 1 G₁ is homeomorphic to the Bowditch boundary of G 2 G₂. 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.