{"paper":{"title":"Asymmetry of $\\ell^{2}$-cohomology via skewed F{\\o}lner geometry","license":"http://creativecommons.org/licenses/by/4.0/","headline":"For finitely generated nilpotent groups, left and right ℓ²-Dirichlet subspaces coincide exactly when the group is virtually abelian.","cross_cats":["math.DS"],"primary_cat":"math.GR","authors_text":"Nachi Avraham-Re'em, Zemer Kosloff","submitted_at":"2026-05-12T16:32:36Z","abstract_excerpt":"We study the two canonical $\\ell^{2}$-Dirichlet structures on a finitely generated group $G$, arising from the left and right regular actions on $\\mathbb{R}^{G}$. Although the left and right regular representations are unitarily equivalent, their $\\ell^{2}$-Dirichlet subspaces of $\\mathbb{R}^{G}$ need not coincide. We prove that for finitely generated nilpotent groups this $\\ell^{2}$-asymmetry is governed by virtual commutativity: $$\\mathcal{D}_{2} \\left(G,\\lambda\\right) = \\mathcal{D}_{2} \\left(G,\\rho \\right) \\quad \\Longleftrightarrow \\quad G \\text{ is virtually abelian}.$$ The proof introduce"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We prove that for finitely generated nilpotent groups this ℓ²-asymmetry is governed by virtual commutativity: D₂(G,λ) = D₂(G,ρ) ⇔ G is virtually abelian.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The groups under study are finitely generated and nilpotent, which supplies the commutator structure and Følner sequence properties needed for left schemes to detect asymmetry (abstract, paragraph on nilpotent case).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"For finitely generated nilpotent groups, left and right ℓ²-Dirichlet structures coincide if and only if the group is virtually abelian, via skewed Følner geometry and left schemes; this yields first asymmetric Bernoulli schemes over amenable groups.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"For finitely generated nilpotent groups, left and right ℓ²-Dirichlet subspaces coincide exactly when the group is virtually abelian.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"f30d16a544843a305d7b9761f210fc585085e1f0ed70b2cd45527222157e7dca"},"source":{"id":"2605.12360","kind":"arxiv","version":2},"verdict":{"id":"a1a6be68-e086-4d3f-8f50-0f19cf3d9719","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T17:24:37.824831Z","strongest_claim":"We prove that for finitely generated nilpotent groups this ℓ²-asymmetry is governed by virtual commutativity: D₂(G,λ) = D₂(G,ρ) ⇔ G is virtually abelian.","one_line_summary":"For finitely generated nilpotent groups, left and right ℓ²-Dirichlet structures coincide if and only if the group is virtually abelian, via skewed Følner geometry and left schemes; this yields first asymmetric Bernoulli schemes over amenable groups.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The groups under study are finitely generated and nilpotent, which supplies the commutator structure and Følner sequence properties needed for left schemes to detect asymmetry (abstract, paragraph on nilpotent case).","pith_extraction_headline":"For finitely generated nilpotent groups, left and right ℓ²-Dirichlet subspaces coincide exactly when the group is virtually abelian."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.12360/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"claim_evidence","ran_at":"2026-05-19T22:41:58.244179Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T10:38:03.418292Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T08:01:19.028571Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T07:37:05.702347Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"54a7e93763b163e00bf708ea667178282f78fd02d3ba496cb192235077b9a7b2"},"references":{"count":26,"sample":[{"doi":"","year":1997,"title":"An introduction to infinite ergodic theory , author=. 1997 , publisher=","work_id":"b70a1317-5f37-4758-87c1-d8af2db98eb9","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Group cohomology, harmonic functions and the first","work_id":"82d5fe5a-1b11-484c-840e-af12febfa039","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Bekka, Bachir and de la Harpe, Pierre and Valette, Alain , title =","work_id":"2b819626-2455-4b0c-a7d3-d02701582dab","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Bernoulli actions of amenable groups with weakly mixing","work_id":"fe4a56cb-0aa5-4644-a995-1dbdc2b2906a","ref_index":4,"cited_arxiv_id":"1808.05991","is_internal_anchor":true},{"doi":"","year":null,"title":"Ergodicity and type of nonsingular","work_id":"9f617ed9-88ec-4e4f-ad07-4485c0191f61","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":26,"snapshot_sha256":"bc4ceeef8e9721744ac7cc78bdad7283a8abddbcd2d68ac861a1d1b2b7a23562","internal_anchors":1},"formal_canon":{"evidence_count":2,"snapshot_sha256":"7d0cd877289277d0b60349bd35d024f023c77188922ddd358a6b567d46054303"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}