Uniform doubling on compact homogeneous spaces with cyclic Lie brackets
Abstract
We establish the uniform doubling property for G-invariant metrics on a wide class of compact Riemannian homogeneous spaces (G/K,g), describing explicitly the volume growth of the metric balls Bg(p,r). This property was previously known only for specific cases, including abelian Lie groups, the Lie group SU(2) and quotients of SU(2)× ℝn. Building on the approach of Eldredge, Gordina and Saloff-Coste for SU(2), we refine and develop geometric and Lie-theoretic tools that allow us to replace the explicit identities of the Milnor basis in SU(2) by a general structural assumption on the metric eigenspace decomposition. In particular, we show that the uniform doubling of (G/K,g) is a consequence of the cyclic bracket condition [mi,mj]=mk, for i,j,k pairwise distinct, a property that naturally generalizes the Milnor structure of SU(2). We apply our results to ℤ2× ℤ2-symmetric spaces, establishing the uniform doubling property for the complete family of G-invariant metrics on several classes of generalized Wallach spaces. For compact homogeneous spaces, we also derive a global Poincaré inequality which holds uniformly for the spaces (G/K,g) under consideration, with a constant controlled by the volume doubling constant of the space.
Related
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BibTeX
@misc{souris2026uniform,
title = {{Uniform doubling on compact homogeneous spaces with cyclic Lie brackets}},
author = {Nikolaos Panagiotis Souris},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06765},
eprint = {2609.06765},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06765},
}
RIS
TY - UNPB
TI - Uniform doubling on compact homogeneous spaces with cyclic Lie brackets
AU - Nikolaos Panagiotis Souris
PY - 2026
DA - 2026/09/06
PB - arXiv
DO - 10.48550/arxiv.2609.06765
UR - https://arxiv.org/abs/2609.06765
AB - We establish the uniform doubling property for G-invariant metrics on a wide class of compact Riemannian homogeneous spaces (G/K,g), describing explicitly the volume growth of the metric balls B_g(p,r). This property was previously known only for specific cases, including abelian Lie groups, the Lie group SU(2) and quotients of SU(2)× ℝ^n. Building on the approach of Eldredge, Gordina and Saloff-Coste for SU(2), we refine and develop geometric and Lie-theoretic tools that allow us to replace the explicit identities of the Milnor basis in SU(2) by a general structural assumption on the metric eigenspace decomposition. In particular, we show that the uniform doubling of (G/K,g) is a consequence of the cyclic bracket condition [m_i,m_j]=m_k, for i,j,k pairwise distinct, a property that naturally generalizes the Milnor structure of SU(2). We apply our results to ℤ_2× ℤ_2-symmetric spaces, establishing the uniform doubling property for the complete family of G-invariant metrics on several classes of generalized Wallach spaces. For compact homogeneous spaces, we also derive a global Poincaré inequality which holds uniformly for the spaces (G/K,g) under consideration, with a constant controlled by the volume doubling constant of the space.
ER -