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The dynamics and geometry of free group endomorphisms
- Publication Year :
- 2020
- Publisher :
- arXiv, 2020.
-
Abstract
- We prove that ascending HNN extensions of free groups are word-hyperbolic if and only if they have no Baumslag-Solitar subgroups. This extends the theorem of Brinkmann that free-by-cyclic groups are word-hyperbolic if and only if they have no free abelian subgroups of rank 2. The paper is split into two independent parts: 1) We study the dynamics of injective nonsurjective endomorphisms of free groups. We prove a canonical structure theorem that initializes the development of improved relative train tracks for endomorphisms; this structure theorem is of independent interest since it makes many open questions about injective endomorphisms tractable. 2) As an application of the structure theorem, we are able to (relatively) combine Brinkmann's theorem with our previous work and obtain the main result stated above. In the final section, we further extend the result to HNN extensions of free groups over free factors.<br />Comment: v1: 59 pages, 6 figures. v2: 60 pages, 6 figures. The existence result in the first part of the paper is now an existence & uniqueness result
- Subjects :
- Pure mathematics
Endomorphism
Rank (linear algebra)
General Mathematics
010102 general mathematics
Group Theory (math.GR)
01 natural sciences
Injective function
Mathematics::Group Theory
Development (topology)
Section (category theory)
20F65, 20E05, 20E06, 20F67
0103 physical sciences
Free group
FOS: Mathematics
010307 mathematical physics
0101 mathematics
Abelian group
Mathematics - Group Theory
Structured program theorem
Mathematics
Subjects
Details
- Database :
- OpenAIRE
- Accession number :
- edsair.doi.dedup.....1aa93a5aed0828006133bfde70ab9f53
- Full Text :
- https://doi.org/10.48550/arxiv.2005.11896