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

We study the two ${\mathrm{\ell }}^2$-Dirichlet structures on a countable group $G$ arising from the left and right regular actions on ${\mathbb{R}}^G$. Although the two regular representations are unitarily equivalent, their ${\mathrm{\ell }}^2$-Dirichlet subspaces of ${\mathbb{R}}^G$ need not coincide. Our main result gives a complete classification of this asymmetry for countable amenable groups: $${\mathcal{D}}_2(G,\lambda )={\mathcal{D}}_2(G,\rho )⟺G\text{ is an FC-group}\mathrm{.}$$ The proof is based on a skewed Følner-geometric mechanism, called a left scheme, combining summability of left boundaries with displacement under a right translation. We develop this mechanism generally, and demonstrate it concretely in the Heisenberg group and amenable wreath products over $\mathbb{Z}$. We also show that this mechanism has a dynamical counterpart in the theory of nonsingular Bernoulli shifts: every countable amenable group that is not an FC-group admits Bernoulli schemes whose left shift is nonsingular, conservative and weakly mixing, whereas the right shift by some element is singular.