Embedding Partial HNN Extensions In Ascending HNN Extensions

TL;DR

This paper proves that partial ascending HNN extensions of free groups can be embedded into full ascending HNN extensions, and further as parabolic subgroups within relatively hyperbolic groups, expanding understanding of their structural relationships.

Contribution

It demonstrates that any partial ascending HNN extension of a free group can be embedded into a full ascending HNN extension, and as a parabolic subgroup in a relatively hyperbolic group, providing new embedding results.

Findings

Partial ascending HNN extensions embed into ascending HNN extensions.

Such embeddings can be realized as parabolic subgroups in relatively hyperbolic groups.

The results generalize the understanding of subgroup embeddings in free groups.

Abstract

We show that any partial ascending HNN extension of a free group embeds in an actual ascending HNN extension of a free group. Moreover, we can ensure that it embeds as the parabolic subgroup of a relatively hyperbolic group.