Quasiisometry invariance of relatively hyperbolic TDLC groups

Motivated by the work of Druţu–Sapir, we introduce a definition of relatively hyperbolic TDLC groups that depends only on their Cayley-Abels graphs and show that this new definition is equivalent to one introduced by Arora–Martínez-Pedroja. Then, we prove that relative hyperbolicity for TDLC groups is invariant under quasiisometry.