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PreprintCC BY 4.0arXiv

Proof of the singularity conjecture for discrete subgroups of SL2(ℝ)

  • Timothée Bénard
Open at sourcePDFDOICite

Abstract

Let μ be a finitely supported probability measure on SL2(ℝ) whose support generates a Zariski-dense discrete subgroup. We prove that its stationary probability measure on ℙ1(ℝ) is singular with respect to the SO(2)-invariant probability measure. The proof compares winding variances on finite covers of a closed hyperbolic surface and uses a topological obstruction.

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Cite

BibTeXRIS
BibTeX
@misc{benard2026proof,
  title = {{Proof of the singularity conjecture for discrete subgroups of $\mathrmSL_2(\mathbbR)$}},
  author = {Timothée Bénard},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.17506},
  eprint = {2609.17506},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.17506},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Proof of the singularity conjecture for discrete subgroups of SL_2(ℝ)
AU  - Timothée Bénard
PY  - 2026
DA  - 2026/09/15
PB  - arXiv
DO  - 10.48550/arxiv.2609.17506
UR  - https://arxiv.org/abs/2609.17506
AB  - Let μ be a finitely supported probability measure on SL_2(ℝ) whose support generates a Zariski-dense discrete subgroup. We prove that its stationary probability measure on ℙ^1(ℝ) is singular with respect to the SO(2)-invariant probability measure. The proof compares winding variances on finite covers of a closed hyperbolic surface and uses a topological obstruction.
ER  -