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

For <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> n≥ 2 , let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>G</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>=</m:mo> <m:mrow> <m:msub> <m:mi>A</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo lspace="0.222em" rspace="0.222em">∗</m:mo> <m:mi mathvariant="normal">⋯</m:mi> <m:mo lspace="0.222em" rspace="0.222em">∗</m:mo> <m:msub> <m:mi>A</m:mi> <m:mi>n</m:mi> </m:msub> </m:mrow> </m:mrow> </m:math> G₁=A₁ Aₙ and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>G</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>=</m:mo> <m:mrow> <m:msub> <m:mi>B</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo lspace="0.222em" rspace="0.222em">∗</m:mo> <m:mi mathvariant="normal">⋯</m:mi> <m:mo lspace="0.222em" rspace="0.222em">∗</m:mo> <m:msub> <m:mi>B</m:mi> <m:mi>n</m:mi> </m:msub> </m:mrow> </m:mrow> </m:math> G₂=B₁ Bₙ where the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>A</m:mi> <m:mi>i</m:mi> </m:msub> </m:math> Aᵢ ’s and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>B</m:mi> <m:mi>i</m:mi> </m:msub> </m:math> Bᵢ ’s are non-elementary relatively hyperbolic groups. Suppose that, for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>1</m:mn> <m:mo>≤</m:mo> <m:mi>i</m:mi> <m:mo>≤</m:mo> <m:mi>n</m:mi> </m:mrow> </m:math> 1≤ i≤ n , the Bowditch boundary of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>A</m:mi> <m:mi>i</m:mi> </m:msub> </m:math> Aᵢ is homeomorphic to the Bowditch boundary of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>B</m:mi> <m:mi>i</m:mi> </m:msub> </m:math> Bᵢ . We show that the Bowditch boundary of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>G</m:mi> <m:mn>1</m:mn> </m:msub> </m:math> G₁ is homeomorphic to the Bowditch boundary of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>G</m:mi> <m:mn>2</m:mn> </m:msub> </m:math> 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.