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If G is defined by the vector $e=(e_1,...,e_{p-1})\\in\\F_p^{p-1}$, the determination of the order of $G_n$ is split into three cases, according as e is non-symmetric, non-constant symmetric, or constant. The formulas that we obtain only depend on p, n, and the rank of the circulant matrix whose first row is e. As a consequence of these formulas, we also obtain the Hausdorff dimension of the closures of all GGS-groups over the p-adic tree."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1108.2289","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2011-08-10T20:59:59Z","cross_cats_sorted":[],"title_canon_sha256":"3d357fa1d19cadb704cd5c63f90e6203b93a1800ef48156b9980dcf4e127ca90","abstract_canon_sha256":"1e8060641c74c46c7ec354a69c943a49029a75699cbc3007e410454182f39870"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:15:38.228675Z","signature_b64":"pxEPS0D2Iv6n6c05B7DbG5Ih6r8gSlZIePyF2wbgrlveim4x8WcBh3xApIUoErxz1AAOe3mnr8hEvWhZ+k5XDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a1e7fe834cb4127f49d2c30445b03a6ef2bf26d5c24515c8e819104d33443567","last_reissued_at":"2026-05-18T04:15:38.227653Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:15:38.227653Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"GGS-groups: order of congruence quotients and Hausdorff dimension","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Amaia Zugadi-Reizabal, Gustavo A. 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