Asymmetry of $\ell^{2}$-cohomology via skewed Følner geometry

Abstract: We study the two canonical $\ell<sup>{2}$-Dirichlet structures on a finitely generated group $G$, arising from the left and right regular actions on $\mathbb{R}<sup>{G}$. Although the left and right regular representations are unitarily equivalent, their $\ell<sup>{2}$-Dirichlet spaces need not coincide as subspaces of $\mathbb{R}<sup>{G}$. We prove that for finitely generated nilpotent groups $G$ this $\ell<sup>{2}$-asymmetry is governed exactly by virtual commutativity: $$\mathcal{D}<em>{2}\left(G,λ\right)=\mathcal{D}</em>{2}\left(G,ρ\right)\quad\Longleftrightarrow\quad G \text{ is virtually abelian}.$$ The proof introduces a skewed Følner-geometric mechanism, called a left scheme, combining summability of left boundaries with displacement under right translation. By refining this mechanism into recurrent left scheme, 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 $\ell<sup>{2}$-asymmetry arises from one-sided commensurable ends rather than from left schemes.