On the positive theory of groups acting on trees

TL;DR

This paper investigates the positive theory of groups acting on trees, demonstrating that with weak small cancellation elements, the positive theory becomes trivial, aligning with that of a free group, and applies to many significant classes of groups.

Contribution

It introduces a uniform method for constructing small cancellation tuples from stable elements, leading to new insights into the positive theory of various complex groups.

Findings

Positive theory is trivial under weak small cancellation conditions.

Applicable to fundamental groups of 3-manifolds, Baumslag-Solitar groups, and most one-relator groups.

Results include quantifier reduction and invariance of positive theory under group extensions.

Abstract

We study the positive theory of groups acting on trees and show that under the presence of weak small cancellation elements, the positive theory of the group is trivial, i.e. coincides with the positive theory of a non-abelian free group. Our results apply to a wide class of groups, including non-virtually solvable fundamental groups of $3$-manifold groups, generalised Baumslag-Solitar groups and almost all one-relator groups and graph products of groups. It follows that groups in the class satisfy a number of algebraic properties: for instance, their verbal subgroups have infinite width and, although some groups in the class are simple, they cannot be boundedly simple. In order to prove these results we describe a uniform way for constructing (weak) small cancellation tuples from (weakly) stable elements. This result of interest in its own is fundamental to obtain corollaries of general nature such as a quantifier reduction for positive sentences or the preservation of the non-trivial positive theory under extensions of groups.