{"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"}