Papers
Topics
Authors
Recent
Assistant
AI Research Assistant
Well-researched responses based on relevant abstracts and paper content.
Custom Instructions Pro
Preferences or requirements that you'd like Emergent Mind to consider when generating responses.
Gemini 2.5 Flash
Gemini 2.5 Flash 134 tok/s
Gemini 2.5 Pro 41 tok/s Pro
GPT-5 Medium 30 tok/s Pro
GPT-5 High 26 tok/s Pro
GPT-4o 64 tok/s Pro
Kimi K2 185 tok/s Pro
GPT OSS 120B 442 tok/s Pro
Claude Sonnet 4.5 37 tok/s Pro
2000 character limit reached

The theory of one-relator groups: history and recent progress (2501.18306v1)

Published 30 Jan 2025 in math.GR and math.HO

Abstract: The theory of one-relator groups is now almost a century old. The authors therefore feel that a comprehensive survey of this fascinating subject is in order, and this document is an attempt at precisely such a survey. This article is divided into two chapters, reflecting the two different phases in the story of one-relator groups. The first chapter, written by the second author, covers the historical development of the theory roughly until the advent of geometric group theory. The second chapter, written by the first author, covers the recent progress in the theory up until the present day. The two chapters can be read independently of one another and have minimal overlap.

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now
Dice Question Streamline Icon: https://streamlinehq.com

Open Problems

  1. Magnus conjecture on isomorphisms of one‑relator groups
  2. General conjugacy problem for one-relator groups
  3. General isomorphism problem for one-relator groups
  4. Decidability of residual finiteness for one-relator groups
  5. Equivalence of Nielsen–Schreier to the axiom of choice
  6. Existence of finitely presented monoids with minimal relators and undecidable word problem
  7. Conjugacy problem for residually nilpotent one-relator groups
  8. Core decision problems for two-relator groups
  9. Recognition of positive one-relator groups
  10. Gersten’s conjecture on hyperbolicity of one-relator groups
  11. Residual finiteness of hyperbolic one-relator groups
  12. Wise’s conjecture on virtual free-by-cyclicity
  13. Chong–Wise conjecture on embedding free-by-cyclic groups
  14. Wilton’s conjecture on groups with uniform negative immersions
  15. Louder–Wilton conjecture on relative hyperbolicity at primitivity rank 2
  16. Mel’nikov’s conjecture on groups whose finite-index subgroups are one-relator
  17. Remeslennikov’s conjecture on profinite rigidity of free groups
  18. Submonoid membership in hyperbolic and torsion one-relator groups
  19. Derived series length and maximal residually solvable quotients for one-relator groups
  20. Magnus two‑relator Nielsen–move conjecture
  21. Decidability of conjugacy and isomorphism problems for one‑relator groups
  22. Decidability of residual finiteness for one‑relator groups
  23. Equivalence of Nielsen–Schreier to the axiom of choice
  24. Conjugacy problem for residually nilpotent one‑relator groups
  25. Submonoid membership problems in hyperbolic and torsion one‑relator groups
  26. Decision problems for two‑relator groups
  27. Gersten’s conjecture on hyperbolic one‑relator groups
  28. Wise’s conjecture on virtually free‑by‑cyclic hyperbolic one‑relator groups
  29. Remeslennikov’s conjecture (profinite rigidity of free groups)
  30. Derived‑series stabilization conjecture for one‑relator groups
  31. Wilton’s conjecture on uniform negative immersions
  32. Louder–Wilton conjecture on relative hyperbolicity (π(w)=2 case)
  33. Undistortion conjecture for cubulated one‑relator groups
  34. Chong–Wise embedding conjecture for free‑by‑cyclic groups
Lightbulb Streamline Icon: https://streamlinehq.com

Continue Learning

We haven't generated follow-up questions for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Generate Now
List To Do Tasks Checklist Streamline Icon: https://streamlinehq.com

Collections

Sign up for free to add this paper to one or more collections.

User Circle Single Streamline Icon: https://streamlinehq.com Sign Up
X Twitter Logo Streamline Icon: https://streamlinehq.com

Tweets

This paper has been mentioned in 1 tweet and received 16 likes.

Upgrade to Pro to view all of the tweets about this paper:

Start a free 7-day Pro trial