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We apply this to the case $M_i$ are w-rigid II$_1$ factors, with $B$ equal to either $\\Bbb C$, to a Cartan subalgebra $A$ in $M_i$, or to a regular hyperfinite II$_1$ subfactor $R$ in $M_i$, to obtain the following type of unique decomposition results, \\`a la Bass-Serre: If $M = (N_1 *_C N_2 *_C ...)^t$, for some $t>0$ and some other simila"},"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/0505589","kind":"arxiv","version":6},"metadata":{"license":"","primary_cat":"math.OA","submitted_at":"2005-05-27T18:41:05Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"6da3c8ca5eea036f12391ddd9c60915d6269ec893d495e4b4664edf19704bdd3","abstract_canon_sha256":"47ea7fe3c272394e05c571d1932a88b747b344f91db9a81f953f0f39bc8ba60b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T15:06:29.996100Z","signature_b64":"QvgAakhmK6Y4PY62VocFgMqY4k8Bj4RlNm24ay5Lia/edoNdEevL5HaI+lYKE3hmCWL2bkQ1FHxbECLUKSupAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bcfd1d2ded04ebb6ec54bef19ce0f34da581e49d54855ead835c6b9bad99a787","last_reissued_at":"2026-07-04T15:06:29.995692Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T15:06:29.995692Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Amalgamated Free Products of $w$-Rigid Factors and Calculation of their Symmetry Groups","license":"","headline":"","cross_cats":["math.GR"],"primary_cat":"math.OA","authors_text":"A. 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