{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:FECBK6CDKAAZHA4UJSMXTMW4RJ","short_pith_number":"pith:FECBK6CD","canonical_record":{"source":{"id":"1101.4055","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2011-01-21T00:36:08Z","cross_cats_sorted":["math.GT"],"title_canon_sha256":"edc179d30d15740ef68e2406447594b6978c56e9d3387212a7888c77f40b8dae","abstract_canon_sha256":"8cb3563186671f3c7b33d749ec6d8533e234bb43bd8ce683df5a73a6e0c0997c"},"schema_version":"1.0"},"canonical_sha256":"290415784350019383944c9979b2dc8a52a1e33e1d0cf518b6665add78dfe8a3","source":{"kind":"arxiv","id":"1101.4055","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1101.4055","created_at":"2026-05-18T02:22:50Z"},{"alias_kind":"arxiv_version","alias_value":"1101.4055v2","created_at":"2026-05-18T02:22:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1101.4055","created_at":"2026-05-18T02:22:50Z"},{"alias_kind":"pith_short_12","alias_value":"FECBK6CDKAAZ","created_at":"2026-05-18T12:26:28Z"},{"alias_kind":"pith_short_16","alias_value":"FECBK6CDKAAZHA4U","created_at":"2026-05-18T12:26:28Z"},{"alias_kind":"pith_short_8","alias_value":"FECBK6CD","created_at":"2026-05-18T12:26:28Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:FECBK6CDKAAZHA4UJSMXTMW4RJ","target":"record","payload":{"canonical_record":{"source":{"id":"1101.4055","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2011-01-21T00:36:08Z","cross_cats_sorted":["math.GT"],"title_canon_sha256":"edc179d30d15740ef68e2406447594b6978c56e9d3387212a7888c77f40b8dae","abstract_canon_sha256":"8cb3563186671f3c7b33d749ec6d8533e234bb43bd8ce683df5a73a6e0c0997c"},"schema_version":"1.0"},"canonical_sha256":"290415784350019383944c9979b2dc8a52a1e33e1d0cf518b6665add78dfe8a3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:22:50.825983Z","signature_b64":"1zSs9aDtGmk2tZotuCrp3MQBmaLPxooHJnXusT+2uHwCgRUfpxt8cthQ/im4UqgC+aK7lH7hg+5mvfeMQfwRCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"290415784350019383944c9979b2dc8a52a1e33e1d0cf518b6665add78dfe8a3","last_reissued_at":"2026-05-18T02:22:50.825525Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:22:50.825525Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1101.4055","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:22:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"q4L6PKX2oemXE5hNt0xirLWvj2+wubFxb0s7JtV+Xzxj6fkkMNp34Pb4wjpNcY9r1Yh+f6GA+a8VNUBNMWNkBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T00:28:47.944045Z"},"content_sha256":"9c169cac452e352dbdd6b1bded2de8ecb05e56d7f2b638a103140cb3b5d15462","schema_version":"1.0","event_id":"sha256:9c169cac452e352dbdd6b1bded2de8ecb05e56d7f2b638a103140cb3b5d15462"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:FECBK6CDKAAZHA4UJSMXTMW4RJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Isometric endomorphisms of free groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GT"],"primary_cat":"math.GR","authors_text":"Alden Walker, Danny Calegari","submitted_at":"2011-01-21T00:36:08Z","abstract_excerpt":"An arbitrary homomorphism between groups is nonincreasing for stable commutator length, and there are infinitely many (injective) homomorphisms between free groups which strictly decrease the stable commutator length of some elements. However, we show in this paper that a random homomorphism between free groups is almost surely an isometry for stable commutator length for every element; in particular, the unit ball in the scl norm of a free group admits an enormous number of exotic isometries.\n  Using similar methods, we show that a random fatgraph in a free group is extremal (i.e. is an absol"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1101.4055","kind":"arxiv","version":2},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:22:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HzyiXPPsd5nKeZDSIozEQcsuG6cWv1Z9slLGi/5Qykf2vqRHDT0DPOHS+OOob+B2D0dWnrDTL6MjJuBsdmSGAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T00:28:47.944621Z"},"content_sha256":"dc0f92fccace0d5238dbb63afb133011aa271ae306656b9301021f722d87128f","schema_version":"1.0","event_id":"sha256:dc0f92fccace0d5238dbb63afb133011aa271ae306656b9301021f722d87128f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FECBK6CDKAAZHA4UJSMXTMW4RJ/bundle.json","state_url":"https://pith.science/pith/FECBK6CDKAAZHA4UJSMXTMW4RJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FECBK6CDKAAZHA4UJSMXTMW4RJ/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-28T00:28:47Z","links":{"resolver":"https://pith.science/pith/FECBK6CDKAAZHA4UJSMXTMW4RJ","bundle":"https://pith.science/pith/FECBK6CDKAAZHA4UJSMXTMW4RJ/bundle.json","state":"https://pith.science/pith/FECBK6CDKAAZHA4UJSMXTMW4RJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FECBK6CDKAAZHA4UJSMXTMW4RJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:FECBK6CDKAAZHA4UJSMXTMW4RJ","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":"8cb3563186671f3c7b33d749ec6d8533e234bb43bd8ce683df5a73a6e0c0997c","cross_cats_sorted":["math.GT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2011-01-21T00:36:08Z","title_canon_sha256":"edc179d30d15740ef68e2406447594b6978c56e9d3387212a7888c77f40b8dae"},"schema_version":"1.0","source":{"id":"1101.4055","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1101.4055","created_at":"2026-05-18T02:22:50Z"},{"alias_kind":"arxiv_version","alias_value":"1101.4055v2","created_at":"2026-05-18T02:22:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1101.4055","created_at":"2026-05-18T02:22:50Z"},{"alias_kind":"pith_short_12","alias_value":"FECBK6CDKAAZ","created_at":"2026-05-18T12:26:28Z"},{"alias_kind":"pith_short_16","alias_value":"FECBK6CDKAAZHA4U","created_at":"2026-05-18T12:26:28Z"},{"alias_kind":"pith_short_8","alias_value":"FECBK6CD","created_at":"2026-05-18T12:26:28Z"}],"graph_snapshots":[{"event_id":"sha256:dc0f92fccace0d5238dbb63afb133011aa271ae306656b9301021f722d87128f","target":"graph","created_at":"2026-05-18T02:22:50Z","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":"An arbitrary homomorphism between groups is nonincreasing for stable commutator length, and there are infinitely many (injective) homomorphisms between free groups which strictly decrease the stable commutator length of some elements. However, we show in this paper that a random homomorphism between free groups is almost surely an isometry for stable commutator length for every element; in particular, the unit ball in the scl norm of a free group admits an enormous number of exotic isometries.\n  Using similar methods, we show that a random fatgraph in a free group is extremal (i.e. is an absol","authors_text":"Alden Walker, Danny Calegari","cross_cats":["math.GT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2011-01-21T00:36:08Z","title":"Isometric endomorphisms of free groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1101.4055","kind":"arxiv","version":2},"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":"sha256:9c169cac452e352dbdd6b1bded2de8ecb05e56d7f2b638a103140cb3b5d15462","target":"record","created_at":"2026-05-18T02:22:50Z","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":"8cb3563186671f3c7b33d749ec6d8533e234bb43bd8ce683df5a73a6e0c0997c","cross_cats_sorted":["math.GT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2011-01-21T00:36:08Z","title_canon_sha256":"edc179d30d15740ef68e2406447594b6978c56e9d3387212a7888c77f40b8dae"},"schema_version":"1.0","source":{"id":"1101.4055","kind":"arxiv","version":2}},"canonical_sha256":"290415784350019383944c9979b2dc8a52a1e33e1d0cf518b6665add78dfe8a3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"290415784350019383944c9979b2dc8a52a1e33e1d0cf518b6665add78dfe8a3","first_computed_at":"2026-05-18T02:22:50.825525Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:22:50.825525Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"1zSs9aDtGmk2tZotuCrp3MQBmaLPxooHJnXusT+2uHwCgRUfpxt8cthQ/im4UqgC+aK7lH7hg+5mvfeMQfwRCg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:22:50.825983Z","signed_message":"canonical_sha256_bytes"},"source_id":"1101.4055","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9c169cac452e352dbdd6b1bded2de8ecb05e56d7f2b638a103140cb3b5d15462","sha256:dc0f92fccace0d5238dbb63afb133011aa271ae306656b9301021f722d87128f"],"state_sha256":"ea96ab1d6d61b47e3e8dc2cbf236b1a8e863849155c78fd33135904c8c265dc8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WkKiNNwu3TGLxNIFseRIizlE7j8OxCi8HMy4qjVCCY9AKcBWZQYv1COgi9BXVkVqIMfDbVkAHFC8wNHWiUdcDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T00:28:47.946875Z","bundle_sha256":"665f40d7202326309a640aed8f10fce89bea9144df685d7c48cac703f31ad2b7"}}