Quasi‐isometry invariance of group splittings over coarse Poincaré duality groups - Margolis, A (Wiley, 2018)
We show that if G is a group of type F P n + 1 Z 2 that is coarsely separated into three essential, coarse disjoint, coarse complementary components by a coarse P D n Z 2 space W , then W is at finite Hausdorff distance from a subgroup H of G ; moreover, G splits over a subgroup commensurable to a subgroup of H . We use this to deduce that splittings of the form G = A ∗ H B , where G is of type F P n + 1 Z 2 and H is a coarse P D n Z 2 group such that both | Comm A ( H ) : H | and | Comm B ( H ) : H | are greater than two, are invariant under quasi‐isometry.