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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"},"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":true,"formal_links_present":true},"canonical_record":{"source":{"id":"2605.12360","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GR","submitted_at":"2026-05-12T16:32:36Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"8fc05fb772e96dd19011af785af347817df814f251f3560896188416f12a84c7","abstract_canon_sha256":"81a20d08970fcf2c268eec2378591d971e551a9aad4118a59ceefe934ce83008"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-20T00:00:43.076060Z","signature_b64":"RUP6cqjtglTJnYDVhVV7Zi2Yf1WnYnxQq1XtwPjdN0kve18CR0lZfopO3gT5vex1a3L7OMgzng+K4z6b772ECA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"046e151fed4ba51e7bf6a8bc442f8558c1bf775268feeb51687c5b5778acfb19","last_reissued_at":"2026-05-20T00:00:43.075263Z","signature_status":"signed_v1","first_computed_at":"2026-05-20T00:00:43.075263Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 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