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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","authors_text":"Nachi Avraham-Re'em, Zemer Kosloff","cross_cats":["math.DS"],"headline":"For finitely generated nilpotent groups, left and right ℓ²-Dirichlet subspaces coincide exactly when the group is virtually abelian.","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GR","submitted_at":"2026-05-12T16:32:36Z","title":"Asymmetry of $\\ell^{2}$-cohomology via skewed F{\\o}lner geometry"},"references":{"count":26,"internal_anchors":1,"resolved_work":26,"sample":[{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":1,"title":"An introduction to infinite ergodic theory , author=. 1997 , publisher=","work_id":"b70a1317-5f37-4758-87c1-d8af2db98eb9","year":1997},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":2,"title":"Group cohomology, harmonic functions and the first","work_id":"82d5fe5a-1b11-484c-840e-af12febfa039","year":null},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":3,"title":"Bekka, Bachir and de la Harpe, Pierre and Valette, Alain , title =","work_id":"2b819626-2455-4b0c-a7d3-d02701582dab","year":null},{"cited_arxiv_id":"1808.05991","doi":"","is_internal_anchor":true,"ref_index":4,"title":"Bernoulli actions of amenable groups with weakly mixing","work_id":"fe4a56cb-0aa5-4644-a995-1dbdc2b2906a","year":null},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":5,"title":"Ergodicity and type of nonsingular","work_id":"9f617ed9-88ec-4e4f-ad07-4485c0191f61","year":null}],"snapshot_sha256":"bc4ceeef8e9721744ac7cc78bdad7283a8abddbcd2d68ac861a1d1b2b7a23562"},"source":{"id":"2605.12360","kind":"arxiv","version":2},"verdict":{"created_at":"2026-05-19T17:24:37.824831Z","id":"a1a6be68-e086-4d3f-8f50-0f19cf3d9719","model_set":{"reader":"grok-4.3"},"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; 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