{"paper":{"title":"On minimal non-sofic and $\\omega$-non-sofic groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"If a minimal non-sofic group has a finitely generated residually finite maximal normal subgroup, then that subgroup is central and the group is a perfect central extension of a finitely generated non-amenable simple group.","cross_cats":[],"primary_cat":"math.GR","authors_text":"K{\\i}van\\c{c} Ersoy","submitted_at":"2026-04-21T07:38:54Z","abstract_excerpt":"We investigate structural properties of non-sofic groups, assuming that such groups exist. We introduce and study two classes: minimal non-sofic groups and $\\omega$-non-sofic groups. For minimal non-sofic groups, we establish strong structural restrictions. In particular, we show that if $G$ is a minimal non-sofic group and $M$ is a finitely generated residually finite maximal normal subgroup of $G$, then $M$ is central and $G$ is a perfect central extension of a finitely generated non-amenable simple group.\n  On the other hand, we show that locally graded non-sofic groups are necessarily $\\om"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"if G is a minimal non-sofic group and M is a finitely generated residually finite maximal normal subgroup of G, then M is central and G is a perfect central extension of a finitely generated non-amenable simple group","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The paper assumes the existence of non-sofic groups and, for the minimal case, the existence of a finitely generated residually finite maximal normal subgroup M whose properties can be analyzed via residual finiteness and centrality (abstract, paragraph on minimal non-sofic groups)","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Assuming non-sofic groups exist, minimal non-sofic groups with a finitely generated residually finite maximal normal subgroup are perfect central extensions of finitely generated non-amenable simple groups, and locally graded non-sofic groups are ω-non-sofic with nontrivial profinite residual chains","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"If a minimal non-sofic group has a finitely generated residually finite maximal normal subgroup, then that subgroup is central and the group is a perfect central extension of a finitely generated non-amenable simple group.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"efa45fb6346e899d4ceafde6f50bc05aaa9c373ab710f7de2e266e12d39bda00"},"source":{"id":"2604.19174","kind":"arxiv","version":3},"verdict":{"id":"27da77aa-38b7-4708-99a1-842381f849ff","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T17:50:49.325347Z","strongest_claim":"if G is a minimal non-sofic group and M is a finitely generated residually finite maximal normal subgroup of G, then M is central and G is a perfect central extension of a finitely generated non-amenable simple group","one_line_summary":"Assuming non-sofic groups exist, minimal non-sofic groups with a finitely generated residually finite maximal normal subgroup are perfect central extensions of finitely generated non-amenable simple groups, and locally graded non-sofic groups are ω-non-sofic with nontrivial profinite residual chains","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The paper assumes the existence of non-sofic groups and, for the minimal case, the existence of a finitely generated residually finite maximal normal subgroup M whose properties can be analyzed via residual finiteness and centrality (abstract, paragraph on minimal non-sofic groups)","pith_extraction_headline":"If a minimal non-sofic group has a finitely generated residually finite maximal normal subgroup, then that subgroup is central and the group is a perfect central extension of a finitely generated non-amenable simple group."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.19174/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":13,"sample":[{"doi":"","year":1999,"title":"M. Gromov. Endomorphisms of symbolic algebraic varieties.J. Eur. Math. Soc. (JEMS), 1(2):109–197, 1999","work_id":"fc075689-a58e-482f-922d-6a86aad64259","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2015,"title":"V. Capraro and M. Lupini.Introduction to Sofic and Hyperlinear Groups and Connes’ Embedding Conjecture. Lecture Notes in Mathematics 2136. Springer, Cham, 2015","work_id":"f04d4a89-fb76-44ec-bf68-1d9bb431b5de","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1937,"title":"B. H. Neumann. Some remarks on infinite groups.J. Lond. Math. Soc., 12:120–127, 1937","work_id":"14fd0205-cb8e-4f0c-8268-5be6e785e574","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2025,"title":"Brescia, K.ıvanç Ersoy, and M","work_id":"a7457245-cfac-4cb5-9f71-2555102360a2","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1989,"title":"S. V. Ivanov and A. Yu. Ol’shanskii. Some applications of graded diagrams in com- binatorial group theory. InGroups, Vol. 2, Proc. Int. Conf., St. Andrews/UK 1989, London Math. Soc. Lecture Note Ser. ","work_id":"97a1c766-eb68-49d3-8f94-6f1aa9c7b574","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":13,"snapshot_sha256":"17159de93ae23dfb57e5b5b2655b3cf4d9cdbeb295b19f13afb3f9e81a32cdc8","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"c85691e520f40948f35e9914eed612067ac03dcca4ebb30f9af1197518be7732"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}