Skip to content
DraconiticResearch
Search Sources API
203,234 studies31 sourcesUpdated 4 min ago
PreprintarXiv licencearXiv

Proof of the positive trace gap conjecture

  • Nikolay Bogachev
Open at sourcePDFDOICite

Abstract

We prove that a lattice Γ in PSL2(ℝ) or PSL2(ℂ) has positive trace gap, meaning that its traces are uniformly separated, if and only if it is derived from an admissible quaternion algebra. For cocompact Fuchsian groups, this proves the positive trace gap conjecture attributed to Sarnak by Geninska and Leuzinger in 2008. The same characterization by quaternion algebras holds if the difference set of traces is not dense. Our method also gives a similar result for lattices in SLd(ℝ), for every d≥3: a lattice Γ<SLd(ℝ) has positive trace gap if and only if all its traces are integers. Equivalently, after conjugation, it has finite index in the norm-one group of an order in a central simple algebra of degree d over ℚ that splits over ℝ. We also discuss spectral consequences and other applications of the results and techniques.

Related

  1. Chern's Conjecture with Constant Cubic Trace
    Huixin Tan, Zizhou Tang, Yuquan Xie +1
    arXivPreprintCC BY-NC-ND 4.0Full textGeometry & topologyarXiv
  2. Local Gan-Gross-Prasad conjecture for unitary groups: Trace formula approach for epsilon dichotomy
    Cheng Chen, Xiaowei Li, Bushi Ren
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  3. The Second Gap and Rigidity in Chern's Conjecture with Constant Cubic Trace
    Jianquan Ge, Huixin Tan, Wenjiao Yan +1
    arXivPreprintCC BY-NC-ND 4.0Full textGeometry & topologyarXiv
  4. Stochastic trace estimation for positive trace-class operators
    Zvonimir Bujanović, Luka Grubišić, Daniel Kressner +1
    arXivPreprintCC BY 4.0Full textNumerical analysis & scientific computingarXiv
  5. A trace lower bound for totally positive units in ℚ(ζp)+ and its application to fusion category
    Yue Meng, Zhiqiang Yu
    arXivPreprintCC0Full textAlgebra & number theoryarXiv

Cite

BibTeXRIS
BibTeX
@misc{bogachev2026proof,
  title = {{Proof of the positive trace gap conjecture}},
  author = {Nikolay Bogachev},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.29033},
  eprint = {2609.29033},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.29033},
}
RIS
TY  - UNPB
TI  - Proof of the positive trace gap conjecture
AU  - Nikolay Bogachev
PY  - 2026
DA  - 2026/09/24
PB  - arXiv
DO  - 10.48550/arxiv.2609.29033
UR  - https://arxiv.org/abs/2609.29033
AB  - We prove that a lattice Γ in PSL_2(ℝ) or PSL_2(ℂ) has positive trace gap, meaning that its traces are uniformly separated, if and only if it is derived from an admissible quaternion algebra. For cocompact Fuchsian groups, this proves the positive trace gap conjecture attributed to Sarnak by Geninska and Leuzinger in 2008. The same characterization by quaternion algebras holds if the difference set of traces is not dense. Our method also gives a similar result for lattices in SL_d(ℝ), for every d≥3: a lattice Γ<SL_d(ℝ) has positive trace gap if and only if all its traces are integers. Equivalently, after conjugation, it has finite index in the norm-one group of an order in a central simple algebra of degree d over ℚ that splits over ℝ. We also discuss spectral consequences and other applications of the results and techniques.
ER  -