Pith Number

pith:ARXBKH7N

pith:2026:ARXBKH7NJOSR467WVC6EIL4FLD

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

Nachi Avraham-Re'em, Zemer Kosloff

For finitely generated nilpotent groups, left and right ℓ²-Dirichlet subspaces coincide exactly when the group is virtually abelian.

arxiv:2605.12360 v2 · 2026-05-12 · math.GR · math.DS

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Claims

C1strongest claim

We prove that for finitely generated nilpotent groups this ℓ²-asymmetry is governed by virtual commutativity: D₂(G,λ) = D₂(G,ρ) ⇔ G is virtually abelian.

C2weakest assumption

The groups under study are finitely generated and nilpotent, which supplies the commutator structure and Følner sequence properties needed for left schemes to detect asymmetry (abstract, paragraph on nilpotent case).

C3one line summary

For finitely generated nilpotent groups, left and right ℓ²-Dirichlet structures coincide if and only if the group is virtually abelian, via skewed Følner geometry and left schemes; this yields first asymmetric Bernoulli schemes over amenable groups.

References

26 extracted · 26 resolved · 1 Pith anchors

[1] An introduction to infinite ergodic theory , author=. 1997 , publisher= 1997

[2] Group cohomology, harmonic functions and the first

[3] Bekka, Bachir and de la Harpe, Pierre and Valette, Alain , title =

[4] Bernoulli actions of amenable groups with weakly mixing · arXiv:1808.05991

[5] Ergodicity and type of nonsingular

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-20T00:00:43.075263Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

046e151fed4ba51e7bf6a8bc442f8558c1bf775268feeb51687c5b5778acfb19

Aliases

arxiv: 2605.12360 · arxiv_version: 2605.12360v2 · doi: 10.48550/arxiv.2605.12360 · pith_short_12: ARXBKH7NJOSR · pith_short_16: ARXBKH7NJOSR467W · pith_short_8: ARXBKH7N

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/ARXBKH7NJOSR467WVC6EIL4FLD \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 046e151fed4ba51e7bf6a8bc442f8558c1bf775268feeb51687c5b5778acfb19

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "81a20d08970fcf2c268eec2378591d971e551a9aad4118a59ceefe934ce83008",
    "cross_cats_sorted": [
      "math.DS"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.GR",
    "submitted_at": "2026-05-12T16:32:36Z",
    "title_canon_sha256": "8fc05fb772e96dd19011af785af347817df814f251f3560896188416f12a84c7"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.12360",
    "kind": "arxiv",
    "version": 2
  }
}