{
  "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 $B_g(p,r)$. This property was previously known only for specific cases, including abelian Lie groups, the Lie group $SU(2)$ and quotients of $SU(2)\\times \\mathbb R^n$. 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 $[\\mathfrak{m}_i,\\mathfrak{m}_j]=\\mathfrak{m}_k$, for $i,j,k$ pairwise distinct, a property that naturally generalizes the Milnor structure of $SU(2)$. We apply our results to $\\mathbb Z_2\\times \\mathbb Z_2$-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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06765",
  "author_total": 1,
  "authors": [
    {
      "name": "Nikolaos Panagiotis Souris"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/630kt3n2bee8x.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06765",
  "fulltext_url": null,
  "id": "630kt3n2bee8x",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.06765",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7212008794",
  "pdf_url": "https://arxiv.org/pdf/2609.06765",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/630kt3n2bee8x.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.06765",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06765"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Uniform doubling on compact homogeneous spaces with cyclic Lie brackets",
  "title_text": "Uniform doubling on compact homogeneous spaces with cyclic Lie brackets",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/630kt3n2bee8x",
  "venue": "arXiv",
  "versions": []
}