Relatively hyperbolic TDLC groups are closed under amalgams and HNN extensions over compact subgroups, and their Bowditch boundaries are determined by the boundaries of the vertex groups.
Bridson and A Haefliger, Metric spaces of nonpositive curvature, Grundlehren der mathematischen Wissenchaften, Vol 319, Springer-Verlag (1999)
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On combination theorems and Bowditch boundaries of relatively hyperbolic TDLC groups
Relatively hyperbolic TDLC groups are closed under amalgams and HNN extensions over compact subgroups, and their Bowditch boundaries are determined by the boundaries of the vertex groups.