Embeddability and universal theory of partially commutative groups

TL;DR

This paper studies embeddability of partially commutative groups via extension graphs, proving decidability results and linking algebraic embeddability to model-theoretic classification and universal equivalence.

Contribution

It introduces an algorithm to decide embeddability via extension graphs and connects this to universal theory classification of partially commutative groups.

Findings

Decidability of the Extension Graph Embedding Problem.

Decidability of embedding for 2-dimensional partially commutative groups.

Algorithm to determine universal equivalence to 2-dimensional groups.

Abstract

The first part of the paper centers in the study of embeddability between partially commutative groups. In [KK], for a finite simplicial graph $\Gamma$, the authors introduce an infinite, locally infinite graph $\Gamma^e$, called the extension graph of $\Gamma$. They show that each finite induced subgraph $\Delta$ of $\Gamma^e$ gives rise to an embedding between the partially commutative groups $G(\Delta)$ and $G({\Gamma})$. Furthermore, it is proven that in many instances the converse also holds. Our first result is the decidability of the Extension Graph Embedding Problem: there is an algorithm that given two finite simplicial graphs {\Delta} and {\Gamma} decides whether or not $\Delta$ is an induced subgraph of $\Gamma^e$. As a corollary we obtain the decidability of the Embedding Problem for 2-dimensional partially commutative groups. In the second part of the paper, we relate the Embedding Problem between partially commutative groups to the model-theoretic question of classification up to universal equivalence. We use our characterisation to transfer algebraic and algorithmic results on embeddability to model-theoretic ones and obtain some rigidity results on the elementary theory of atomic pc groups as well as to deduce the existence of an algorithm to decide if an arbitrary pc group is universally equivalent to a 2-dimensional one.