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PreprintarXiv licencearXiv

The Robust Thompson's Conjecture for Alternating Groups

  • Nathan Keller
  • Noam Lifshitz
  • Avichai Marmor
  • Ohad Sheinfeld
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Abstract

We show that there exists a constant ε> 0 such that if C is a conjugacy class in An of size at least |An|1−ε, then An ∖ {1} ⊆ C2. This proves the robust version of Thompson's conjecture for the alternating group, conjectured by Shalev (Annals of Math., 2009). Our proof combines character bounds and combinatorial cancellation techniques with the recent theory of hypercontractivity for functions over symmetric groups.

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Cite

BibTeXRIS
BibTeX
@misc{keller2026robust,
  title = {{The Robust Thompson's Conjecture for Alternating Groups}},
  author = {Nathan Keller and Noam Lifshitz and Avichai Marmor and Ohad Sheinfeld},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.31453},
  eprint = {2609.31453},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.31453},
}
RIS
TY  - UNPB
TI  - The Robust Thompson's Conjecture for Alternating Groups
AU  - Nathan Keller
AU  - Noam Lifshitz
AU  - Avichai Marmor
AU  - Ohad Sheinfeld
PY  - 2026
DA  - 2026/09/25
PB  - arXiv
DO  - 10.48550/arxiv.2609.31453
UR  - https://arxiv.org/abs/2609.31453
AB  - We show that there exists a constant ε> 0 such that if C is a conjugacy class in A_n of size at least |A_n|^1−ε, then A_n ∖ {1} ⊆ C^2. This proves the robust version of Thompson's conjecture for the alternating group, conjectured by Shalev (Annals of Math., 2009). Our proof combines character bounds and combinatorial cancellation techniques with the recent theory of hypercontractivity for functions over symmetric groups.
ER  -