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Among the group theoretic properties it is proved that they are free-by-(torsion-free poly -procyclic) and if non-abelian do not have a finitely generated non-trivial normal subgroup of infinite index. Furthermore it is shown that every 2 generated pro-$p$ group in the class $\\mathcal{L}$ is either free pro-$p$ "},"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":"1107.2331","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2011-07-12T15:57:56Z","cross_cats_sorted":[],"title_canon_sha256":"99e272f8b3ae872aa42b0e8841ba4901d5b47aff02dd499e1dc55afc7a1cd128","abstract_canon_sha256":"60c68f2c66ddfa3fa053fdd240f1c5e4310a4b9ec7beaea19dfdcd06592d039f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:18:24.801372Z","signature_b64":"TCw7WjsLZRKablkBlPBDsQ97BicZvYfwuXQ+eIlX7zMWE2ARXKPKuPiQnSOXT3+/drejCZqKfxrYKZHWRoA0BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1c71f3d2453c9122d4b37afb0ce71f73a483b86186e2db653246249da0371700","last_reissued_at":"2026-05-18T04:18:24.800719Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:18:24.800719Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On pro-$p$ analogues of limit groups via extensions of centralizers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Dessislava H. 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