A global shadow lemma for relatively Morse groups in higher rank

Dongryul M. Kim; Hee Oh

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arXiv · 2606.19779

A global shadow lemma for relatively Morse groups 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. In this paper, 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. This extends the global shadow lemma of Stratmann-Velani for geometrically finite real hyperbolic groups. As applications, we obtain uniform local estimates for Patterson-Sullivan measures, and we give sufficient conditions under which these measures agree, up to scale, with the Hausdorff measure defined by the associated visual quasi-metric.

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BibTeX RIS

Dongryul M. Kim, Hee Oh. 2026-06-18. A global shadow lemma for relatively Morse groups in higher rank. https://arxiv.org/abs/2606.19779

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