Asymmetry of $\ell^{2}$-cohomology via skewed Følner geometry
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arXiv
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| Natura: | Preprint |
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2026
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| author | Avraham-Re'em, Nachi Kosloff, Zemer |
| author_facet | Avraham-Re'em, Nachi Kosloff, Zemer |
| contents | 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: $$\mathcal{D}_{2} \left(G,λ\right) = \mathcal{D}_{2} \left(G,ρ\right) \quad \Longleftrightarrow \quad G \text{ is virtually abelian}.$$ The proof introduces a skewed Følner-geometric mechanism, called a \emph{left scheme}, combining summability of left boundaries with displacement under right translation. By refining this mechanism into \emph{recurrent left schemes}, we further show that every non-virtually abelian finitely generated nilpotent group admits Bernoulli schemes whose left shift is nonsingular and weakly mixing whereas the right shift is singular. These are the first constructions of such Bernoulli schemes over amenable groups. In addition to nilpotent groups, our techniques are robust enough to cover all amenable wreath products over $\mathbb{Z}$ and solvable Baumslag--Solitar groups. We also classify the virtually cyclic case, where $\ell^{2}$-asymmetry arises from one-sided commensurated ends rather than from left schemes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_12360 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Asymmetry of $\ell^{2}$-cohomology via skewed Følner geometry Avraham-Re'em, Nachi Kosloff, Zemer Group Theory Dynamical Systems Primary 20F18, 20F65, 20F69, Secondary 37A20, 20J06, 37A40, 43A07 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: $$\mathcal{D}_{2} \left(G,λ\right) = \mathcal{D}_{2} \left(G,ρ\right) \quad \Longleftrightarrow \quad G \text{ is virtually abelian}.$$ The proof introduces a skewed Følner-geometric mechanism, called a \emph{left scheme}, combining summability of left boundaries with displacement under right translation. By refining this mechanism into \emph{recurrent left schemes}, we further show that every non-virtually abelian finitely generated nilpotent group admits Bernoulli schemes whose left shift is nonsingular and weakly mixing whereas the right shift is singular. These are the first constructions of such Bernoulli schemes over amenable groups. In addition to nilpotent groups, our techniques are robust enough to cover all amenable wreath products over $\mathbb{Z}$ and solvable Baumslag--Solitar groups. We also classify the virtually cyclic case, where $\ell^{2}$-asymmetry arises from one-sided commensurated ends rather than from left schemes. |
| title | Asymmetry of $\ell^{2}$-cohomology via skewed Følner geometry |
| topic | Group Theory Dynamical Systems Primary 20F18, 20F65, 20F69, Secondary 37A20, 20J06, 37A40, 43A07 |
| url | https://arxiv.org/abs/2605.12360 |