Normalising graphs of groups

TL;DR

This paper introduces a partial normalization technique for graphs of groups that preserves the fundamental group, providing new tools for analyzing finitely generated virtually free groups and their properties.

Contribution

It presents a novel partial normalization method for graphs of groups that aids in studying virtually free groups and their classifications.

Findings

Derived an inequality for edges in Stallings decompositions

Proved equivalence of conditions for a group to be 'large'

Classified virtually free groups of rank 2 up to isomorphism

Abstract

We discuss a partial normalisation of a finite graph of finite groups $(\Gamma(-), X)$ which leaves invariant the fundamental group. In conjunction with an easy graph-theoretic result, this provides a flexible and rather useful tool in the study of finitely generated virtually free groups. Applications discussed here include (i) an important inequality for the number of edges in a Stallings decomposition $\Gamma \cong \pi_1(\Gamma(-), X)$ of a finitely generated virtually free group, (ii) the proof of equivalence of a number of conditions for such a group to be `large', as well as (iii) the classification up to isomorphism of virtually free groups of (free) rank $2$. We also discuss some number-theoretic consequences of the last result.