Elementary subgroups of virtually free groups

TL;DR

This paper characterizes elementary subgroups of finitely generated virtually free groups, providing a description, an algorithm for decision problems, and insights into elementary embeddings in such groups.

Contribution

It offers a new description of elementary subgroups, an algorithm to decide their elementary nature, and proves that elementary embeddings are automorphisms in this context.

Findings

Elementary subgroups of finitely generated free groups are free factors.

An algorithm decides if a subgroup is $orall ext{-}orall ext{-}orall$-elementary.

Elementary embeddings of equationally noetherian groups are automorphisms.

Abstract

We give a description of elementary subgroups (in the sense of first-order logic) of finitely generated virtually free groups. In particular, we recover the fact that elementary subgroups of finitely generated free groups are free factors. Moreover, we give an algorithm that takes as input a finite presentation of a virtually free group $G$ and a finite subset $X$ of $G$, and decides if the subgroup of $G$ generated by $X$ is $\exists\forall\exists$-elementary. We also prove that every elementary embedding of an equationally noetherian group into itself is an automorphism.