Quasi-Isometry Invariance of Group Splittings over Coarse Poincaré Duality Groups

Abstract

We show that if $G$ is a group of type $F{P}_{n+1}^{{\mathbb{Z}}_2}$ that is coarsely separated into three essential, coarse disjoint, coarse complementary components by a coarse $P{D}_n^{{\mathbb{Z}}_2}$ 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 $G=A{\ast }_{H}B$, where $G$ is of type $F{P}_{n+1}^{{\mathbb{Z}}_2}$ and $H$ is a coarse $P{D}_n^{{\mathbb{Z}}_2}$ group such that both $|{\mathrm{Comm}}_A(H):H|$ and $|{\mathrm{Comm}}_B(H):H|$ are greater than two, are invariant under quasi-isometry.

46 pages

References in corpus (1)

• Quasi-isometries Between Groups with Two-Ended Splittings

Cited by in corpus (1)

• The geometry of groups containing almost normal subgroups