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These include examples with nonabelian free subgroups, examples which do not admit faithful actions by $C^2$ diffeomorphisms on $1$--manifolds, examples which do not admit faithful actions by $PL$ homeomorphisms on an interval, and examples which are not finitely presented. We thus answer questions due to M."},"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":"1704.03112","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2017-04-11T01:50:32Z","cross_cats_sorted":["math.DS","math.GT"],"title_canon_sha256":"5fdb4342451eac819fbdce6c4a292ac50ae1b7a794d511ec312caf7c790c2f35","abstract_canon_sha256":"c14de100d89b637cf5c1422085fce315a2f7f56bf770867de71ead19fd98f0bb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:55:43.317042Z","signature_b64":"gDeOT3DGv/INgYs2y1W0RnNC6UaldcfmwPRqYnUrOWPWjX4RmE4Foq7DE4gfK6k9P3dMMM1YkL8fhSFR16w3AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"734a94713e922d50337c0961c1f2fd60671e54e272a5b109f64bb17ad25189e9","last_reissued_at":"2026-05-17T23:55:43.316412Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:55:43.316412Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Two-chains and square roots of Thompson's group $F$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS","math.GT"],"primary_cat":"math.GR","authors_text":"Thomas Koberda, Yash Lodha","submitted_at":"2017-04-11T01:50:32Z","abstract_excerpt":"We study two--generated subgroups $\\langle f,g\\rangle<\\mathrm{Homeo}^+(I)$ such that $\\langle f^2,g^2\\rangle$ is isomorphic to Thompson's group $F$, and such that the supports of $f$ and $g$ form a chain of two intervals. We show that this class contains uncountably many isomorphism types. These include examples with nonabelian free subgroups, examples which do not admit faithful actions by $C^2$ diffeomorphisms on $1$--manifolds, examples which do not admit faithful actions by $PL$ homeomorphisms on an interval, and examples which are not finitely presented. 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