Onto extensions of free groups

TL;DR

This paper introduces and studies onto extensions of subgroups in free groups, exploring their properties, closure operators, and contrasting them with algebraic extensions, revealing limitations for dual notions.

Contribution

It defines onto extensions in free groups, investigates their properties and closure operators, and demonstrates the triviality of dual into extensions, expanding understanding of subgroup extensions.

Findings

Onto extensions are a broader class than algebraic extensions.

Closure operators for onto extensions are characterized and analyzed.

Dual into extensions are shown to be trivial, limiting Takahasi-type results.

Abstract

An extension of subgroups $H\leqslant K\leqslant F_A$ of the free group of rank $|A|=r\geqslant 2$ is called onto when, for every ambient free basis $A'$, the Stallings graph $\Gamma_{A'}(K)$ is a quotient of $\Gamma_{A'}(H)$. Algebraic extensions are onto and the converse implication was conjectured by Miasnikov-Ventura-Weil, and resolved in the negative, first by Parzanchevski-Puder for rank $r=2$, and recently by Kolodner for general rank. In this note we study properties of this new type of extension among free groups (as well as the fully onto variant), and investigate their corresponding closure operators. Interestingly, the natural attempt for a dual notion -- into extensions -- becomes trivial, making a Takahasi type theorem not possible in this setting.