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We prove that there is a coarse group $F_{\\mathfrak{M}} (X, \\mathcal{E})\\in \\mathfrak{M}$ such that $(X, \\mathcal{E}) $ is a subspace of $F_{\\mathfrak{M}} (X, \\mathcal{E})$, $X$ generates $F_{\\mathfrak{M}} (X, \\mathcal{E})$ and every coarse mapping $(X, \\mathcal{E}) \\longrightarrow (G, \\mathcal{E}^{\\prime}) $ where $G\\in\\mathfrak{M}$, $(G, \\ma"},"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":"1803.10504","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2018-03-28T10:09:04Z","cross_cats_sorted":[],"title_canon_sha256":"01becf7ec70a03ef205cc2bc8ede3af89c27a5b108331df3208eea3b9ad56661","abstract_canon_sha256":"7d59a06763d2f2559f2da7ebb319a4ac2832c2c0a188f824243282259415899b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:19:55.123620Z","signature_b64":"RZIKbPrbXaDJsE9Y4wWw83SdFo2vZswgcwjGJY42gOYgmLShk2aeMPCdjnoM6QFkAs/s66SsHnheSpe/z7i8CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"928408951b33ffcba3c3d3607a9d31e6bfe71bc79044ee04bb8e68d086fa64a0","last_reissued_at":"2026-05-18T00:19:55.122924Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:19:55.122924Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Free coarse groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GN","authors_text":"Igor Protasov, Ksenia Protasova","submitted_at":"2018-03-28T10:09:04Z","abstract_excerpt":"A coarse group is a group endowed with a coarse structure so that the group multiplication and inversion are coarse mappings. 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