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We identify the closure of the outer space $P\\mathcal{O}(G,\\{G_1,\\dots,G_k\\})$ for the axes topology with the space of projective minimal, \\emph{very small} $(G,\\{G_1,\\dots,G_k\\})$-trees, i.e. trees whose arc stabilizers are either trivial, or cyclic, closed under taking roots, and not conjugate into any of the $G_i$'s, and whose tripod stabilizers are trivial. Its topological dimension is equal to $3N+2k-4$, and the boundary has dim"},"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":"1408.0543","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2014-08-03T21:00:28Z","cross_cats_sorted":["math.GT"],"title_canon_sha256":"7e08d99631dbb8c2c11e95b41d92250793a5b72ae95a9d410a5ce0ddf7906150","abstract_canon_sha256":"6156ce39d7507c0e62d805f28913f48b85af73110c55ba51d704e5ad442b4933"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:16:26.746340Z","signature_b64":"t0RLUb93zzM8juoj+pG6VZ6wukDM3g1SZO4DFi2duLyGU/Ze4aqCbuX7/jbENZ1czzOPxG8f3grRirA9ydNTAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dda0219351173b7f831b994b9eaf68f6e567e9150c1e3eb092620d00194cf7e7","last_reissued_at":"2026-05-18T01:16:26.745814Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:16:26.745814Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The boundary of the outer space of a free product","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GT"],"primary_cat":"math.GR","authors_text":"Camille Horbez","submitted_at":"2014-08-03T21:00:28Z","abstract_excerpt":"Let $G$ be a countable group that splits as a free product of groups of the form $G=G_1\\ast\\dots\\ast G_k\\ast F_N$, where $F_N$ is a finitely generated free group. We identify the closure of the outer space $P\\mathcal{O}(G,\\{G_1,\\dots,G_k\\})$ for the axes topology with the space of projective minimal, \\emph{very small} $(G,\\{G_1,\\dots,G_k\\})$-trees, i.e. trees whose arc stabilizers are either trivial, or cyclic, closed under taking roots, and not conjugate into any of the $G_i$'s, and whose tripod stabilizers are trivial. 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