Asymmetry of ℓ^2-cohomology via skewed Følner geometry

Abstract

We study the two ℓ²-Dirichlet structures on a countable group G arising from the left and right regular actions on ℝ^G. Although the two regular representations are unitarily equivalent, their ℓ²-Dirichlet subspaces of ℝ^G need not coincide. Our main result gives a complete classification of this asymmetry for countable amenable groups: 𝒟₂(G,λ)=𝒟₂(G,ρ) Longleftrightarrow G is an FC−group. 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 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.

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BibTeX

@misc{avrahamreem2026asymmetry,
  title = {{Asymmetry of $\ell^2$-cohomology via skewed Følner geometry}},
  author = {Nachi Avraham-Re'em and Zemer Kosloff},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2605.12360},
  eprint = {2605.12360},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.12360},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Asymmetry of ℓ^2-cohomology via skewed Følner geometry
AU  - Nachi Avraham-Re'em
AU  - Zemer Kosloff
PY  - 2026
DA  - 2026/09/22
PB  - arXiv
DO  - 10.48550/arxiv.2605.12360
UR  - https://arxiv.org/abs/2605.12360
AB  - We study the two ℓ^2-Dirichlet structures on a countable group G arising from the left and right regular actions on ℝ^G. Although the two regular representations are unitarily equivalent, their ℓ^2-Dirichlet subspaces of ℝ^G need not coincide. Our main result gives a complete classification of this asymmetry for countable amenable groups: 𝒟_2(G,λ)=𝒟_2(G,ρ) Longleftrightarrow G is an FC−group. 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 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.
ER  -

Asymmetry of ℓ²-cohomology via skewed Følner geometry

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