{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:A6KTUMYDU3WVTRCCOFPDYHY6R2","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"e5568d4947ad0cc9ea5bf07724cc2e3d7c873eb10fde29d5c5ee0f34f12ea1ed","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2017-01-16T18:00:10Z","title_canon_sha256":"a2585165e4a5edd034d49825d4f16e96916ff9eb111a5765922536d54cac1118"},"schema_version":"1.0","source":{"id":"1701.04374","kind":"arxiv","version":5}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1701.04374","created_at":"2026-05-17T23:56:10Z"},{"alias_kind":"arxiv_version","alias_value":"1701.04374v5","created_at":"2026-05-17T23:56:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1701.04374","created_at":"2026-05-17T23:56:10Z"},{"alias_kind":"pith_short_12","alias_value":"A6KTUMYDU3WV","created_at":"2026-05-18T12:31:05Z"},{"alias_kind":"pith_short_16","alias_value":"A6KTUMYDU3WVTRCC","created_at":"2026-05-18T12:31:05Z"},{"alias_kind":"pith_short_8","alias_value":"A6KTUMYD","created_at":"2026-05-18T12:31:05Z"}],"graph_snapshots":[{"event_id":"sha256:afc2fce639df758bb5d236459d1ea5755dd3cdd6eaeed61110db78d93404fdce","target":"graph","created_at":"2026-05-17T23:56:10Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Let $G$ be an infinite group and let $X$ be a finite generating set for $G$ such that the growth series of $G$ with respect to $X$ is a rational function; in this case $G$ is said to have rational growth with respect to $X$. In this paper a result on sizes of spheres (or balls) in the Cayley graph $\\Gamma(G,X)$ is obtained: namely, the size of the sphere of radius $n$ is bounded above and below by positive constant multiples of $n^\\alpha \\lambda^n$ for some integer $\\alpha \\geq 0$ and some $\\lambda \\geq 1$.\n  As an application of this result, a calculation of degree of commutativity (d. c.) is","authors_text":"Motiejus Valiunas","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2017-01-16T18:00:10Z","title":"Rational growth and degree of commutativity of graph products"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.04374","kind":"arxiv","version":5},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:75222c8f257095272b355083a4eaee0ba3a3f6415fafa2a7cf698dc6abec7728","target":"record","created_at":"2026-05-17T23:56:10Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"e5568d4947ad0cc9ea5bf07724cc2e3d7c873eb10fde29d5c5ee0f34f12ea1ed","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2017-01-16T18:00:10Z","title_canon_sha256":"a2585165e4a5edd034d49825d4f16e96916ff9eb111a5765922536d54cac1118"},"schema_version":"1.0","source":{"id":"1701.04374","kind":"arxiv","version":5}},"canonical_sha256":"07953a3303a6ed59c442715e3c1f1e8eb9bceb9c867afcfc981f7b25ed73825a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"07953a3303a6ed59c442715e3c1f1e8eb9bceb9c867afcfc981f7b25ed73825a","first_computed_at":"2026-05-17T23:56:10.630871Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:56:10.630871Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"+Q1l2wKFK6p3qXp6GqoGY1/sd61b3LqJHNiZR4VroI9as9rnPPdcvSdy6V4SozXpBXXBtG0bxfCsL1HUR845DA==","signature_status":"signed_v1","signed_at":"2026-05-17T23:56:10.631552Z","signed_message":"canonical_sha256_bytes"},"source_id":"1701.04374","source_kind":"arxiv","source_version":5}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:75222c8f257095272b355083a4eaee0ba3a3f6415fafa2a7cf698dc6abec7728","sha256:afc2fce639df758bb5d236459d1ea5755dd3cdd6eaeed61110db78d93404fdce"],"state_sha256":"578a03f2970717bb38224eccebcf93ea16a165a74c69416269ae9f6a2db16f92"}