Pith Number

pith:4YRRCVUM

pith:2026:4YRRCVUMZZNGW3OF2RMEY3C3QY

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On Pappus and Anosov Representations of the Modular Group

Richard Evan Schwartz

The Barbot component of discrete faithful representations of the modular group into Isom(SL3(R)/SO(3)) is homeomorphic to R² × [0, ∞).

arxiv:2605.15317 v1 · 2026-05-14 · math.GT

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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.

Claims

C1strongest claim

DFR has a component B, the Barbot component, that is homeomorphic to R² × [0,∞). The boundary of B parametrizes the Pappus representations and the interior consists of Anosov representations.

C2weakest assumption

The space DFR of discrete faithful representations that send the order-2 generator to an isometry with a unique fixed point is non-empty and admits a well-defined connected component B whose topology can be analyzed by the methods of the paper.

C3one line summary

The Barbot component of discrete faithful representations of the modular group into Isom(SL_3(R)/SO(3)) is homeomorphic to R² × [0,∞), with Pappus representations on the boundary and Anosov representations in the interior.

References

13 extracted · 13 resolved · 0 Pith anchors

[1] Hence tr( ρ(σ3σ2σ3σ2)) is a smooth function on the smooth part of R

[2] τ (r2 1r2) = 64 (1 − c2)2(1 − d2)

[3] Putting everything together, we see that ( c1,d 1) and (c2,d 2) give representations that are conjugate in Isom( X) only if they lie in the same θ4-orbit

[4] When b ∈ [1 + √ 2, ∞ ) we have a ∈ (0, ∞ )

[5] See [ BL V, Eq

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-20T00:00:52.348685Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

e62311568cce5a6b6dc5d4584c6c5b860c6a77d58bac98543b12c8baf25cd002

Aliases

arxiv: 2605.15317 · arxiv_version: 2605.15317v1 · doi: 10.48550/arxiv.2605.15317 · pith_short_12: 4YRRCVUMZZNG · pith_short_16: 4YRRCVUMZZNGW3OF · pith_short_8: 4YRRCVUM

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/4YRRCVUMZZNGW3OF2RMEY3C3QY \
  | 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: e62311568cce5a6b6dc5d4584c6c5b860c6a77d58bac98543b12c8baf25cd002

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "d4a0f0875d5ec9eaf55cc11d3549b06e7c4115f74b525c792a20b18291f0b3c5",
    "cross_cats_sorted": [],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.GT",
    "submitted_at": "2026-05-14T18:31:01Z",
    "title_canon_sha256": "3ff3758b589074adc08b293f9ba9d2acf004fa5684603eee92b2c9bc34718cbe"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.15317",
    "kind": "arxiv",
    "version": 1
  }
}