Asymmetry of $\ell^{2}$-cohomology via skewed F{\o}lner geometry

TL;DR

This paper investigates the asymmetry in $ ext{ell}^2$-cohomology structures on finitely generated groups, revealing new geometric mechanisms and constructing novel Bernoulli schemes with specific shift properties.

Contribution

It introduces a skewed F{46}lner-geometric mechanism and constructs Bernoulli schemes with asymmetric shift properties over amenable groups.

Findings

Asymmetry in $ ext{ell}^2$-Dirichlet structures is characterized by virtual abelianness.

Introduces left schemes and recurrent left schemes to analyze group asymmetry.

Constructs Bernoulli schemes with nonsingular and weakly mixing left shifts but singular right shifts.

Abstract

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 introduces a skewed F{\o}lner-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.