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We prove that for every torsion free lattice $\\Gamma\\subset {\\rm Isom} X$ any homology class in $H_1(\\Gamma\\backslash X,\\mathbb F_2)$ has a representative cycle of total length $o_X({\\rm Vol}(\\Gamma\\backslash X))$. As an application we show that $\\dim_{\\mathbb F_2} H_1(\\Gamma\\backslash X,\\mathbb F_2)=o_X({\\rm Vol}(\\Gamma\\backslash X)).$"},"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":"1801.09283","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2018-01-28T20:39:54Z","cross_cats_sorted":[],"title_canon_sha256":"a69ea0b7a0618a5026d1ace9bbe80f5530170bc70eae904b9cde1ce74a597c7d","abstract_canon_sha256":"9070ae04ec26cccaa331025e8d32a31dec764facbef8d377b569cc45eae8376f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:24:57.959141Z","signature_b64":"cNmYTB6631rAPROcD8vRILh+deOHZNWaZ1+cN55zuNbY+tVfgvuQejaM/ejbxPQ/tPlPhxHNVqpQh+Tif09+BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fff3e1ef9a6db3e47bfb7684d3fea45a555d59c5d9f6f9f93be0ed10a7dff0a8","last_reissued_at":"2026-05-18T00:24:57.958352Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:24:57.958352Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Growth of mod$-2$ homology in higher rank locally symmetric spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Mikolaj Fraczyk","submitted_at":"2018-01-28T20:39:54Z","abstract_excerpt":"Let $X$ be a higher rank symmetric space or a Bruhat-Tits building of dimension at least $2$ such that the isometry group of $X$ has property $(T)$. We prove that for every torsion free lattice $\\Gamma\\subset {\\rm Isom} X$ any homology class in $H_1(\\Gamma\\backslash X,\\mathbb F_2)$ has a representative cycle of total length $o_X({\\rm Vol}(\\Gamma\\backslash X))$. As an application we show that $\\dim_{\\mathbb F_2} H_1(\\Gamma\\backslash X,\\mathbb F_2)=o_X({\\rm Vol}(\\Gamma\\backslash X)).$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.09283","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1801.09283","created_at":"2026-05-18T00:24:57.958492+00:00"},{"alias_kind":"arxiv_version","alias_value":"1801.09283v1","created_at":"2026-05-18T00:24:57.958492+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1801.09283","created_at":"2026-05-18T00:24:57.958492+00:00"},{"alias_kind":"pith_short_12","alias_value":"77Z6D342NWZ6","created_at":"2026-05-18T12:32:11.075285+00:00"},{"alias_kind":"pith_short_16","alias_value":"77Z6D342NWZ6I673","created_at":"2026-05-18T12:32:11.075285+00:00"},{"alias_kind":"pith_short_8","alias_value":"77Z6D342","created_at":"2026-05-18T12:32:11.075285+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/77Z6D342NWZ6I673O2CNH7VELJ","json":"https://pith.science/pith/77Z6D342NWZ6I673O2CNH7VELJ.json","graph_json":"https://pith.science/api/pith-number/77Z6D342NWZ6I673O2CNH7VELJ/graph.json","events_json":"https://pith.science/api/pith-number/77Z6D342NWZ6I673O2CNH7VELJ/events.json","paper":"https://pith.science/paper/77Z6D342"},"agent_actions":{"view_html":"https://pith.science/pith/77Z6D342NWZ6I673O2CNH7VELJ","download_json":"https://pith.science/pith/77Z6D342NWZ6I673O2CNH7VELJ.json","view_paper":"https://pith.science/paper/77Z6D342","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1801.09283&json=true","fetch_graph":"https://pith.science/api/pith-number/77Z6D342NWZ6I673O2CNH7VELJ/graph.json","fetch_events":"https://pith.science/api/pith-number/77Z6D342NWZ6I673O2CNH7VELJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/77Z6D342NWZ6I673O2CNH7VELJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/77Z6D342NWZ6I673O2CNH7VELJ/action/storage_attestation","attest_author":"https://pith.science/pith/77Z6D342NWZ6I673O2CNH7VELJ/action/author_attestation","sign_citation":"https://pith.science/pith/77Z6D342NWZ6I673O2CNH7VELJ/action/citation_signature","submit_replication":"https://pith.science/pith/77Z6D342NWZ6I673O2CNH7VELJ/action/replication_record"}},"created_at":"2026-05-18T00:24:57.958492+00:00","updated_at":"2026-05-18T00:24:57.958492+00:00"}