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Iwip endomorphisms of free groups and fixed points of graph selfmaps

  • Peng Wang and Qiang Zhang ORCID logo EMAIL logo
Published/Copyright: December 3, 2024
Journal of Group Theory
From the journal Journal of Group Theory Volume 28 Issue 4

Abstract

In a paper from 2011, Jiang, Wang and Zhang studied the fixed points and fixed subgroups of selfmaps on a connected finite graph or a connected compact hyperbolic surface 𝑋. In particular, for any selfmap f : X → X , they proved that a certain quantity defined in terms of the characteristic chr ⁡ ( f , F ) and the index ind ⁡ ( f , F ) of a fixed point class 𝐅 of 𝑓 is bounded below by 2 ⁢ χ ⁢ ( X ) , where χ ⁢ ( X ) is the Euler characteristic of 𝑋. In this paper, we give a sufficient condition for when equality holds and hence we partially answer a question of Jiang. We do this by studying iwip outer endomorphisms of free groups acting on stable trees.

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 12271385 and 12471066

Funding statement: The authors are partially supported by NSFC (No. 12271385 and 12471066), the Shaanxi Fundamental Science Research Project for Mathematics and Physics (No. 23JSY027), and the Fundamental Research Funds for the Central Universities.

Acknowledgements

The authors would like to thank the anonymous referee very much for his/her valuable and detailed comments that helped to greatly improve our earlier draft, especially for pointing out an error in the proof of the main theorems, and providing detailed corrections and Example 6.2.

  1. Communicated by: Evgenii I. Khukhro

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Received: 2023-12-03
Revised: 2024-11-09
Published Online: 2024-12-03
Published in Print: 2025-07-01

© 2024 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. On the kernel of actions on asymptotic cones
  3. CAT(0) cube complexes and asymptotically rigid mapping class groups
  4. Iwip endomorphisms of free groups and fixed points of graph selfmaps
  5. Space of orders with finite Cantor–Bendixson rank
  6. Lifting subgroups of PSL2 to SL2 over local fields
  7. Regular 3-polytopes of order 2𝑛𝑝
  8. On the number of tuples of group elements satisfying a first-order formula
  9. Exponent-critical groups
  10. On soluble groups in which commutators have prime power order
  11. A character theoretic criterion for Fitting height
  12. Hilbert divisors and degrees of irreducible Brauer characters
  13. Character triples and relative defect zero characters
https://doi.org/10.1515/jgth-2023-0283

Articles in the same Issue

  1. Frontmatter
  2. On the kernel of actions on asymptotic cones
  3. CAT(0) cube complexes and asymptotically rigid mapping class groups
  4. Iwip endomorphisms of free groups and fixed points of graph selfmaps
  5. Space of orders with finite Cantor–Bendixson rank
  6. Lifting subgroups of PSL2 to SL2 over local fields
  7. Regular 3-polytopes of order 2𝑛𝑝
  8. On the number of tuples of group elements satisfying a first-order formula
  9. Exponent-critical groups
  10. On soluble groups in which commutators have prime power order
  11. A character theoretic criterion for Fitting height
  12. Hilbert divisors and degrees of irreducible Brauer characters
  13. Character triples and relative defect zero characters
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