{"paper":{"title":"On the finitely generated Hausdorff spectrum of spinal groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GT"],"primary_cat":"math.GR","authors_text":"Elisabeth Fink","submitted_at":"2013-01-29T16:45:45Z","abstract_excerpt":"We study the finitely generated Hausdorff spectrum of spinal automorphism groups acting on rooted trees. Given any $\\alpha \\in [0,1]$, we construct a branch group $G_\\alpha$ such that $G_\\alpha$ has a finitely generated subgroup $H$ where $H$ has Hausdorff dimension $\\alpha$ in $G$. Using results by Barnea, Shalev and Klopsch we further deduce that the finitely generated Hausdorff spectrum of this group $G_\\alpha$ contains $\\mathcal{L}_\\alpha \\cup ([0, 1] \\cap \\mathcal{L})$, where $\\mathcal{L}$ is a countable subset of $\\mathbb{Q}$ and $\\mathcal{L}_\\alpha$ is a certain set of countably many ir"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1301.6977","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}