Coarse Alexander duality for pairs and applications

TL;DR

This paper extends coarse Alexander duality to a relative setting, providing new tools for analyzing coarse embeddings and their complements, with applications to the structure of certain 3-manifold groups.

Contribution

It introduces a relative version of coarse Alexander duality satisfying the Eilenberg-Steenrod axioms, enabling detailed analysis of coarse embeddings and their complements.

Findings

Established a relative coarse Alexander duality theorem.

Proved the existence of a 3-manifold group that is virtually Kleinian but not Kleinian.

Analyzed the structure of complements of certain 2-complex embeddings in 3-manifolds.

Abstract

For a group $G$ (of type $F$) acting properly on a coarse Poincar\'{e} duality space $X$, Kapovich-Kleiner introduced a coarse version of Alexander duality between $G$ and its complement in $X$. More precisely, the cohomology of $G$ with group ring coefficients is dual to a certain \v{C}ech homology group of the family of increasing neighborhoods of a $G$-orbit in $X$. This duality applies more generally to coarse embeddings of certain contractible simplicial complexes into coarse $PD(n)$ spaces. In this paper we introduce a relative version of this \v{C}ech homology that satisfies the Eilenberg-Steenrod Exactness Axiom, and we prove a relative version of coarse Alexander duality. As an application we provide a detailed proof of the following result, first stated by Kapovich-Kleiner. Given a $2$-complex formed by gluing $k$ halfplanes along their boundary lines and a coarse embedding into a contractible $3$-manifold, the complement consists of $k$ deep components that are arranged cyclically in a pattern called a Jordan cycle. We use the Jordan cycle as an invariant in proving the existence of a $3$-manifold group that is virtually Kleinian but not itself Kleinian.