# Quasi-Isometry Invariance of Group Splittings over Coarse Poincar\'e Duality Groups

## TL;DR

This paper proves that certain group splittings over coarse Poincaré duality groups are invariant under quasi-isometry, linking geometric properties to algebraic group decompositions.

## Contribution

It establishes the quasi-isometry invariance of group splittings over coarse Poincaré duality groups, extending geometric group theory understanding.

## Key findings

- Splittings over coarse Poincaré duality groups are invariant under quasi-isometry.
- Subgroups involved in splittings are at finite Hausdorff distance from coarse Poincaré duality subgroups.
- Groups of type FP_{n+1}^{Z_2} split over subgroups commensurable to coarse Poincaré duality groups.

## Abstract

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