Stability of Universal Equivalence of Groups under Free Constructions

TL;DR

This paper proves that the property of universal equivalence in pregroups is preserved when passing to their universal groups, with implications for free products with amalgamation and HNN-extensions.

Contribution

It establishes that universal equivalence of pregroups extends to their universal groups, providing new insights into the stability of this property under free constructions.

Findings

Universal equivalence of pregroups extends to universal groups.

Applications to free products with amalgamation.

Applications to HNN-extensions.

Abstract

In 1971 J. Stallings introduced a generalisation of amalgamated products of groups -- called a pregroup, which is a particular kind of a partial group. He defined the universal group U(P) of a pregroup P to be a universal object (in the sense of category theory) extending the partial operations on P to group operations on U(P). This turns out to be a versatile and convenient generalisation of classical group constructions: HNN-extensions and amalgamated products. In this respect the following general question arises. Which properties of pregroups, or relations between pregroups, carry over to the respective universal groups? In this paper it is proved that universal equivalence of pregroups extends to universal equivalence of universal groups. Applications to free products with amalgamation and HNN-extensions are then described.