Building Groups From Restricted Diagrams of Groups

TL;DR

This paper investigates how restrictions on vertex groups in a graph of groups influence the realizability of a group as a fundamental group, with applications to constructing manifolds from finite building blocks.

Contribution

It introduces new restrictions on realizable groups based on vertex group classes and applies these findings to manifold construction problems.

Findings

Restrictions limit possible fundamental groups in graphs of groups

Topological applications to manifold construction from finite building blocks

Characterization of groups realizable under specific vertex group constraints

Abstract

We consider the problem of realizing a group as the fundamental group of a graph of groups where the vertex groups are restricted to certain classes (for example, coming from a certain finite list of groups, or having bounded geometric rank). We show how this places restrictions on the possible groups that can be realized and we give a topological application of our results to the problem of constructing manifolds from a finite set of "building blocks".