{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:IWY3SKPGGCSH6YAC4XJ2NRKPVB","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":"3baad457d085464940ea4f1c945ecb23d1c956b68d44dbc0c8cbfa412a91476b","cross_cats_sorted":["math.DG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-11-25T11:13:55Z","title_canon_sha256":"1c0b445647251694c45d8c255475c2a6ff6476d0d81043bf738eaac70830a3c9"},"schema_version":"1.0","source":{"id":"1411.6817","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1411.6817","created_at":"2026-05-18T01:36:34Z"},{"alias_kind":"arxiv_version","alias_value":"1411.6817v3","created_at":"2026-05-18T01:36:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1411.6817","created_at":"2026-05-18T01:36:34Z"},{"alias_kind":"pith_short_12","alias_value":"IWY3SKPGGCSH","created_at":"2026-05-18T12:28:33Z"},{"alias_kind":"pith_short_16","alias_value":"IWY3SKPGGCSH6YAC","created_at":"2026-05-18T12:28:33Z"},{"alias_kind":"pith_short_8","alias_value":"IWY3SKPG","created_at":"2026-05-18T12:28:33Z"}],"graph_snapshots":[{"event_id":"sha256:ae06ae6ef34ce8a9fb9a3410a09d3478b0343d3111b7efe327a2f0ee42f5c2f6","target":"graph","created_at":"2026-05-18T01:36:34Z","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 $\\Gamma$ be a (non-elementary) convex co-compact group of isometries of a pinched Hadamard manifold $X$. We show that a normal subgroup $\\Gamma_0$ has critical exponent equal to the critical exponent of $\\Gamma$ if and only if $\\Gamma / \\Gamma_0$ is amenable. We prove a similar result for the exponential growth rate of closed geodesics on $X / \\Gamma$. These statements are analogues of classical results of Kesten for random walks on groups and of Brooks for the spectrum of the Laplacian on covers of Riemannian manifolds.","authors_text":"Rhiannon Dougall, Richard Sharp","cross_cats":["math.DG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-11-25T11:13:55Z","title":"Amenability, Critical Exponents of Subgroups and Growth of Closed Geodesics"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.6817","kind":"arxiv","version":3},"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:a44fdba2e16a28a706497f713d4add6ddf9f2aec29418ae256abb4e3a414e1ec","target":"record","created_at":"2026-05-18T01:36:34Z","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":"3baad457d085464940ea4f1c945ecb23d1c956b68d44dbc0c8cbfa412a91476b","cross_cats_sorted":["math.DG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-11-25T11:13:55Z","title_canon_sha256":"1c0b445647251694c45d8c255475c2a6ff6476d0d81043bf738eaac70830a3c9"},"schema_version":"1.0","source":{"id":"1411.6817","kind":"arxiv","version":3}},"canonical_sha256":"45b1b929e630a47f6002e5d3a6c54fa84cdee47f8085a880466aeac643da6a2b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"45b1b929e630a47f6002e5d3a6c54fa84cdee47f8085a880466aeac643da6a2b","first_computed_at":"2026-05-18T01:36:34.176964Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:36:34.176964Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"1Qo9ot5UFri2wjanMyLEjXOKIolwUCkLnDoywKzm3bpyckoM8C9FYyhPeRgBDftbBJfZwkXEZlK1r+6ngWAqCA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:36:34.177733Z","signed_message":"canonical_sha256_bytes"},"source_id":"1411.6817","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a44fdba2e16a28a706497f713d4add6ddf9f2aec29418ae256abb4e3a414e1ec","sha256:ae06ae6ef34ce8a9fb9a3410a09d3478b0343d3111b7efe327a2f0ee42f5c2f6"],"state_sha256":"89ccf1680352baa7cacb83b55f24a033d6925a5e82d350aee4ebd586e8ebfe2b"}