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However, for every $m \\geq 3$ and every $1 \\leq k \\leq n-1$, there exist a subgroup $H$ of $F_n$ of rank $m$ and a retract $R$ of $F_n$ of rank $k$ such that $H \\cap R$ is not a retract of $H$. This gives a complete answer to a question of Bergman.\n  Furthermore, we provide positive evidence for the inertia conjecture of Dicks and Ventura. More precisely, we prove that $\\textrm{rk}(H \\cap \\textrm{Fix}(S)) \\l"},"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":"1902.02378","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2019-02-06T19:46:55Z","cross_cats_sorted":[],"title_canon_sha256":"f63eb0b07ae4483a4484800aa1a8bfcb5d672a58a1d344a1a28a15c29b3ffd08","abstract_canon_sha256":"0417c1a81e98d6ce3b2e8a54c5e2e98b34373f0d24d05356184e30762ca1c789"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:54:33.602221Z","signature_b64":"9toHKSNvaSfnSbunO3ngS2MBWlyeTT0GlsuRev09eZ7KfQC5muZzjEt8MTlYpF6f4l7ktmZG3Ga5uDHYKWWGBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f061ce417a244afd2b92461b5a5ef05cb96e1687bcf078a6d04262017ee74d69","last_reissued_at":"2026-05-17T23:54:33.601481Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:54:33.601481Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Retracts of free groups and a question of Bergman","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Ilir Snopce, Pavel Zalesskii, Slobodan Tanushevski","submitted_at":"2019-02-06T19:46:55Z","abstract_excerpt":"Let $F_n$ be a free group of finite rank $n \\geq 2$. We prove that if $H$ is a subgroup of $F_n$ with $\\textrm{rk}(H)=2$ and $R$ is a retract of $F_n$, then $H \\cap R$ is a retract of $H$. However, for every $m \\geq 3$ and every $1 \\leq k \\leq n-1$, there exist a subgroup $H$ of $F_n$ of rank $m$ and a retract $R$ of $F_n$ of rank $k$ such that $H \\cap R$ is not a retract of $H$. This gives a complete answer to a question of Bergman.\n  Furthermore, we provide positive evidence for the inertia conjecture of Dicks and Ventura. More precisely, we prove that $\\textrm{rk}(H \\cap \\textrm{Fix}(S)) \\l"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1902.02378","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":"1902.02378","created_at":"2026-05-17T23:54:33.601591+00:00"},{"alias_kind":"arxiv_version","alias_value":"1902.02378v1","created_at":"2026-05-17T23:54:33.601591+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1902.02378","created_at":"2026-05-17T23:54:33.601591+00:00"},{"alias_kind":"pith_short_12","alias_value":"6BQ44QL2ERFP","created_at":"2026-05-18T12:33:10.108867+00:00"},{"alias_kind":"pith_short_16","alias_value":"6BQ44QL2ERFP2K4S","created_at":"2026-05-18T12:33:10.108867+00:00"},{"alias_kind":"pith_short_8","alias_value":"6BQ44QL2","created_at":"2026-05-18T12:33:10.108867+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"1906.11689","citing_title":"On verbally closed subgroups of free solvable groups","ref_index":7,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/6BQ44QL2ERFP2K4SIYNVUXXQLS","json":"https://pith.science/pith/6BQ44QL2ERFP2K4SIYNVUXXQLS.json","graph_json":"https://pith.science/api/pith-number/6BQ44QL2ERFP2K4SIYNVUXXQLS/graph.json","events_json":"https://pith.science/api/pith-number/6BQ44QL2ERFP2K4SIYNVUXXQLS/events.json","paper":"https://pith.science/paper/6BQ44QL2"},"agent_actions":{"view_html":"https://pith.science/pith/6BQ44QL2ERFP2K4SIYNVUXXQLS","download_json":"https://pith.science/pith/6BQ44QL2ERFP2K4SIYNVUXXQLS.json","view_paper":"https://pith.science/paper/6BQ44QL2","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1902.02378&json=true","fetch_graph":"https://pith.science/api/pith-number/6BQ44QL2ERFP2K4SIYNVUXXQLS/graph.json","fetch_events":"https://pith.science/api/pith-number/6BQ44QL2ERFP2K4SIYNVUXXQLS/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/6BQ44QL2ERFP2K4SIYNVUXXQLS/action/timestamp_anchor","attest_storage":"https://pith.science/pith/6BQ44QL2ERFP2K4SIYNVUXXQLS/action/storage_attestation","attest_author":"https://pith.science/pith/6BQ44QL2ERFP2K4SIYNVUXXQLS/action/author_attestation","sign_citation":"https://pith.science/pith/6BQ44QL2ERFP2K4SIYNVUXXQLS/action/citation_signature","submit_replication":"https://pith.science/pith/6BQ44QL2ERFP2K4SIYNVUXXQLS/action/replication_record"}},"created_at":"2026-05-17T23:54:33.601591+00:00","updated_at":"2026-05-17T23:54:33.601591+00:00"}