Pith Number

pith:Q7JS3ORQ

pith:2015:Q7JS3ORQXWOJKDLOPGZMAXXNAQ

not attested not anchored not stored refs pending

l2-Betti numbers of discrete and non-discrete groups

Roman Sauer

arxiv:1509.05234 v2 · 2015-09-17 · math.AT · math.GR

Open paper page JSON Open Graph Bundle Merged state Verified badge What is a Pith Number?

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{Q7JS3ORQXWOJKDLOPGZMAXXNAQ}

Prints a linked badge after your title and injects PDF metadata. Compiles on arXiv. Learn more · Embed verified badge

Record completeness

1 Bitcoin timestamp

2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications open

✓ Portable graph bundle live · download bundle · merged state

The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Receipt and verification

First computed	2026-05-18T00:51:03.362502Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

87d32dba30bd9c950d6e79b2c05eed0410cdef5dc4ebd23551fd9f8cc27f4693

Aliases

arxiv: 1509.05234 · arxiv_version: 1509.05234v2 · doi: 10.48550/arxiv.1509.05234 · pith_short_12: Q7JS3ORQXWOJ · pith_short_16: Q7JS3ORQXWOJKDLO · pith_short_8: Q7JS3ORQ

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/Q7JS3ORQXWOJKDLOPGZMAXXNAQ \
  | 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: 87d32dba30bd9c950d6e79b2c05eed0410cdef5dc4ebd23551fd9f8cc27f4693

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "515b7d7e674dd71b9f96836c292ea8f69a82ef826c58bdb80cfef0623612a5b2",
    "cross_cats_sorted": [
      "math.GR"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.AT",
    "submitted_at": "2015-09-17T12:39:18Z",
    "title_canon_sha256": "ab64e08ea36f4eb3a271a1ece0cf1eb54cddc5dd3ce23c4b3de7a3a51b4a345d"
  },
  "schema_version": "1.0",
  "source": {
    "id": "1509.05234",
    "kind": "arxiv",
    "version": 2
  }
}