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Global shadow estimates and Hausdorff measure criteria in higher rank

  • Dongryul M. Kim
  • Hee Oh
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Abstract

Patterson--Sullivan measures encode the distribution of orbits of discrete group actions near the boundary. We prove a global shadow lemma for Patterson--Sullivan measures associated to relatively Morse subgroups of higher-rank semisimple Lie groups. The estimate is uniform for shadows centered at arbitrary points in a Gromov model, including points deep in the cuspidal part, and expresses the contribution of each cusp in terms of peripheral growth. This extends the global shadow lemma of Stratmann--Velani for geometrically finite real hyperbolic groups. As applications, we establish local doubling and reverse doubling. Under Zariski density, we give necessary and sufficient criteria for Patterson--Sullivan measures to agree with the associated Hausdorff measures and to be Ahlfors regular. For symmetric linear forms, these criteria hold without Zariski density. We verify the criteria explicitly for cusped Hitchin representations and real-hyperbolic self-joinings.

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Cite

BibTeXRIS
BibTeX
@misc{kim2026global,
  title = {{Global shadow estimates and Hausdorff measure criteria in higher rank}},
  author = {Dongryul M. Kim and Hee Oh},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2606.19779},
  eprint = {2606.19779},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.19779},
}
RIS
TY  - UNPB
TI  - Global shadow estimates and Hausdorff measure criteria in higher rank
AU  - Dongryul M. Kim
AU  - Hee Oh
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2606.19779
UR  - https://arxiv.org/abs/2606.19779
AB  - Patterson--Sullivan measures encode the distribution of orbits of discrete group actions near the boundary. We prove a global shadow lemma for Patterson--Sullivan measures associated to relatively Morse subgroups of higher-rank semisimple Lie groups. The estimate is uniform for shadows centered at arbitrary points in a Gromov model, including points deep in the cuspidal part, and expresses the contribution of each cusp in terms of peripheral growth. This extends the global shadow lemma of Stratmann--Velani for geometrically finite real hyperbolic groups. As applications, we establish local doubling and reverse doubling. Under Zariski density, we give necessary and sufficient criteria for Patterson--Sullivan measures to agree with the associated Hausdorff measures and to be Ahlfors regular. For symmetric linear forms, these criteria hold without Zariski density. We verify the criteria explicitly for cusped Hitchin representations and real-hyperbolic self-joinings.
ER  -