Groups of cohomological codimension one

TL;DR

This paper explores the structure of groups with specific cohomological properties, showing how certain almost normal subgroups influence the group's decomposition into a graph of groups, and investigates related subgroup properties.

Contribution

It establishes a new connection between cohomological dimension, almost normal subgroups, and the decomposition of groups into graphs of groups.

Findings

If $H$ is almost normal in $G$ with vcd($G$) = vcd($H$) + 1, then $G$ is a fundamental group of a graph of groups with vertex and edge groups commensurable to $H$.

The paper investigates properties of almost normal subgroups in one-relator groups and duality groups.

Provides structural insights into groups with cohomological codimension one.

Abstract

We show that if $H$ is an almost normal subgroup of $G$ such that both $H$ and $G$ are of type $VFP$ and vcd($G$) = vcd($H$) + 1, then $G$ is the fundamental group of a graph of groups in which all vertex and edge groups are commensurable to $H$. We also investigate almost normal subgroups of one-relator groups and duality groups.