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We define the critical exponent $\\delta(\\mu)$ of any discrete invariant random subgroup $\\mu$ of the locally compact group $G$ and show that $\\delta(\\mu) > \\frac{d}{2}$ in general and that $\\delta(\\mu) = d$ if $\\mu$ is of divergence type. Whenever $G$ is a rank-one simple Lie group with Kazhdan's property $(T)$ it follows that an ergodic invariant random subgroup of divergence type "},"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":"1804.02995","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2018-04-09T14:05:14Z","cross_cats_sorted":["math.DS","math.GT"],"title_canon_sha256":"84f751f9d43d2c3ab8d652a450d8bab26ebcc20ca57dc71d03930550aedc28ab","abstract_canon_sha256":"d3cbbea257004c571603e552475117d39509833f38a31a87f9db6bc6b024583b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:04:37.448462Z","signature_b64":"NkMp0yp/yPMx5IYTXMg0en07RklmNoMpxZKtBr5Y31EG+obeHyoUxqc5bYG/+PVjgwfHkECz2v1Ji4z1T4hhDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1328ddb194864995d6f967f20558fe9b99435424a7d996df59cbcd339e3fecd2","last_reissued_at":"2026-05-18T00:04:37.447970Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:04:37.447970Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Critical exponents of invariant random subgroups in negative curvature","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS","math.GT"],"primary_cat":"math.GR","authors_text":"Arie Levit, Ilya Gekhtman","submitted_at":"2018-04-09T14:05:14Z","abstract_excerpt":"Let $X$ be a proper geodesic Gromov hyperbolic metric space and let $G$ be a cocompact group of isometries of $X$ admitting a uniform lattice. 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Whenever $G$ is a rank-one simple Lie group with Kazhdan's property $(T)$ it follows that an ergodic invariant random subgroup of divergence type "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.02995","kind":"arxiv","version":4},"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":"1804.02995","created_at":"2026-05-18T00:04:37.448041+00:00"},{"alias_kind":"arxiv_version","alias_value":"1804.02995v4","created_at":"2026-05-18T00:04:37.448041+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1804.02995","created_at":"2026-05-18T00:04:37.448041+00:00"},{"alias_kind":"pith_short_12","alias_value":"CMUN3MMUQZEZ","created_at":"2026-05-18T12:32:16.446611+00:00"},{"alias_kind":"pith_short_16","alias_value":"CMUN3MMUQZEZLVXZ","created_at":"2026-05-18T12:32:16.446611+00:00"},{"alias_kind":"pith_short_8","alias_value":"CMUN3MMU","created_at":"2026-05-18T12:32:16.446611+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/CMUN3MMUQZEZLVXZM7ZAKWH6TO","json":"https://pith.science/pith/CMUN3MMUQZEZLVXZM7ZAKWH6TO.json","graph_json":"https://pith.science/api/pith-number/CMUN3MMUQZEZLVXZM7ZAKWH6TO/graph.json","events_json":"https://pith.science/api/pith-number/CMUN3MMUQZEZLVXZM7ZAKWH6TO/events.json","paper":"https://pith.science/paper/CMUN3MMU"},"agent_actions":{"view_html":"https://pith.science/pith/CMUN3MMUQZEZLVXZM7ZAKWH6TO","download_json":"https://pith.science/pith/CMUN3MMUQZEZLVXZM7ZAKWH6TO.json","view_paper":"https://pith.science/paper/CMUN3MMU","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1804.02995&json=true","fetch_graph":"https://pith.science/api/pith-number/CMUN3MMUQZEZLVXZM7ZAKWH6TO/graph.json","fetch_events":"https://pith.science/api/pith-number/CMUN3MMUQZEZLVXZM7ZAKWH6TO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CMUN3MMUQZEZLVXZM7ZAKWH6TO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CMUN3MMUQZEZLVXZM7ZAKWH6TO/action/storage_attestation","attest_author":"https://pith.science/pith/CMUN3MMUQZEZLVXZM7ZAKWH6TO/action/author_attestation","sign_citation":"https://pith.science/pith/CMUN3MMUQZEZLVXZM7ZAKWH6TO/action/citation_signature","submit_replication":"https://pith.science/pith/CMUN3MMUQZEZLVXZM7ZAKWH6TO/action/replication_record"}},"created_at":"2026-05-18T00:04:37.448041+00:00","updated_at":"2026-05-18T00:04:37.448041+00:00"}