Ping-pong for basis-conjugating HNN-extension of free group

2025/09/23 by Vasily Ionin, Ionin, Vasily
Mathematics · #20E06 #20F36 #Advanced Operator Algebra Research #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2509.19635

openalex publication_date 2025/09/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We isolate a tractable class of HNN-extensions of a free group, namely, multiple HNN-extensions by basis-conjugating embeddings. For this class, we construct a normal form and establish a practical version of the ping-pong lemma that provides verifiable sufficient conditions for a set of elements to generate a free subgroup. We then apply these results to the pure braid group P_n+1, exploiting its well-known decomposition as a semidirect product of free groups. Our approach yields new families of free subgroups within the first two factors F_n \rtimes F_n-1 of this decomposition.

Citations

• Embedding Partial HNN Extensions In Ascending HNN Extensions
• Two-generator subgroups of the pure braid group
• On Basis-Conjugating Automorphisms of Free Groups

Related

• The Conjugacy Problem for ascending HNN-extensions of free groups
• Embedding Partial HNN Extensions In Ascending HNN Extensions
• Two-generator subgroups of the pure braid group
• HNN extensions and embedding theorems for groups
• HNN extensions of free groups with equal associated subgroups of finite index: polynomial time word problem