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We explain how these semidirect product structures arise quite naturally from fibrations, and observe that, in each case, the action of the braid group ${\\cal B}_n$ on the free group $F$ is classical. We prove that, for each of the semidirect products, the group of automorphisms which leave invariant the normal subgroup $F$ is small: namely, ${\\rm Out}(A(B_n),F)$ has order 2, and ${\\rm Out}(A(D_n"},"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":"math/0210438","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.GR","submitted_at":"2002-10-29T09:47:23Z","cross_cats_sorted":["math.GT"],"title_canon_sha256":"7635a99821e4689ddd5a82317627400d940078385556b02626dcf5bd1599f377","abstract_canon_sha256":"59c7dbcd3ce2f1918fe2d2001cd1434198c48f431ec063f0f60141d841be5219"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T14:36:40.931169Z","signature_b64":"RlJDgy08CsTFgCCqF7nYsnNa4DlUH7V79U3BRfgpvvwn9OjqguzAHDJm5NI8tHMF8Lda1zxQr3FjiWolsYbRAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9c93d413a735f6149d83417bdf39f360fd90ed3a944b71ce2e9241cf088ab996","last_reissued_at":"2026-07-04T14:36:40.930765Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T14:36:40.930765Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Artin groups of type B and D","license":"","headline":"","cross_cats":["math.GT"],"primary_cat":"math.GR","authors_text":"J. 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