{"paper":{"title":"Model-theoretic Tameness in finite extensions of groups","license":"http://creativecommons.org/licenses/by/4.0/","headline":"There exists an ω-stable group whose finite-index extensions and subgroups interpret any countable first-order structure.","cross_cats":["math.GR"],"primary_cat":"math.LO","authors_text":"Saharon Shelah, Yatir Halevi","submitted_at":"2026-05-14T05:12:44Z","abstract_excerpt":"It is shown that finite-index extensions and finite-index subgroups of $\\omega$-stable groups can be model-theoretically wild. More precisely, there exists an $\\omega$-stable group $G$ such that any given countable first-order structure in a finite language is interpretable both in some finite-index extension of $G$ and in some finite-index subgroup of $G$."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"there exists an ω-stable group G such that any given countable first-order structure in a finite language is interpretable both in some finite-index extension of G and in some finite-index subgroup of G.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The existence of a specific ω-stable group G whose finite-index extensions and subgroups allow interpretation of arbitrary countable structures, which depends on the details of the construction provided in the paper.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"There exists an ω-stable group G such that every countable structure in a finite language is interpretable in some finite-index extension and some finite-index subgroup of G.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"There exists an ω-stable group whose finite-index extensions and subgroups interpret any countable first-order structure.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"9a798c219dbb9a3bb4aed4809ac9465a8ef106f892dc9c21f01a6629c55cad5c"},"source":{"id":"2605.14390","kind":"arxiv","version":1},"verdict":{"id":"fc0a573b-add9-4208-a81a-012a26bfd9c7","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T01:57:55.543389Z","strongest_claim":"there exists an ω-stable group G such that any given countable first-order structure in a finite language is interpretable both in some finite-index extension of G and in some finite-index subgroup of G.","one_line_summary":"There exists an ω-stable group G such that every countable structure in a finite language is interpretable in some finite-index extension and some finite-index subgroup of G.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The existence of a specific ω-stable group G whose finite-index extensions and subgroups allow interpretation of arbitrary countable structures, which depends on the details of the construction provided in the paper.","pith_extraction_headline":"There exists an ω-stable group whose finite-index extensions and subgroups interpret any countable first-order structure."},"references":{"count":19,"sample":[{"doi":"","year":1985,"title":"John T. Baldwin. Some notes on stable groups. In The model theory of groups ( N otre D ame, IN , 1985--1987) , volume 11 of Notre Dame Math. Lectures , pages 100--116. Univ. Notre Dame Press, Notre Da","work_id":"052206cb-2b19-4388-921c-b512bf80b5e6","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1981,"title":"A. Baudisch. Subgroups of semifree groups. Acta Math. Acad. Sci. Hungar. , 38(1-4):19--28, 1981","work_id":"63094b9a-d91c-4cc1-a365-ef081698b2e1","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2012,"title":"I. M. Chiswell. Ordering graph products of groups. Internat. J. Algebra Comput. , 22(4):1250037, 14, 2012","work_id":"27b56264-8bc3-425f-a9de-52a64a9aeb6d","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2025,"title":"First-order aspects of artin groups","work_id":"9d5c023d-cd41-425f-a2fc-a2165816fa96","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2016,"title":"Ordered groups and topology , volume 176 of Graduate Studies in Mathematics","work_id":"af4c9426-be7c-416c-a0fd-36eddb60105b","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":19,"snapshot_sha256":"b4e17b47982c25d9eff7e383fc867a1460bc318f2b75616f35bebc2aa3b435bd","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}