{"paper":{"title":"Thick embeddings of graphs into symmetric spaces via coarse geometry","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.GR","math.MG"],"primary_cat":"math.GT","authors_text":"Benjamin Barrett, David Hume, Elia Portnoy, Larry Guth","submitted_at":"2021-12-10T02:53:51Z","abstract_excerpt":"We prove estimates for the optimal volume of thick embeddings of finite graphs into symmetric spaces, generalising results of Kolmogorov-Barzdin and Gromov-Guth for embeddings into Euclidean spaces. We distinguish two very different behaviours depending on the rank of the non-compact factor. For rank at least 2, we construct thick embeddings of $N$-vertex graphs with volume $CN\\ln(1+N)$ and prove that this is optimal. For rank at most $1$ we prove lower bounds of the form $cN^a$ for some (explicit) $a>1$ which depends on the dimension of the Euclidean factor and the conformal dimension of the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2112.05305","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2112.05305/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}