{"paper":{"title":"On Pappus and Anosov Representations of the Modular Group","license":"http://creativecommons.org/licenses/by/4.0/","headline":"The Barbot component of discrete faithful representations of the modular group into Isom(SL3(R)/SO(3)) is homeomorphic to R² × [0, ∞).","cross_cats":[],"primary_cat":"math.GT","authors_text":"Richard Evan Schwartz","submitted_at":"2026-05-14T18:31:01Z","abstract_excerpt":"Let $X=SL_3(\\R)/SO(3)$. Let $\\cal DFR$ be the space of discrete faithful representations of the modular group into ${\\rm Isom\\/}(X)$ which map the order $2$ generator to an isometry with a unique fixed point. In this paper, we prove that $\\cal DFR$ has a component $\\cal B$, the so-called Barbot component, that is homeomorphic to $\\R^2 \\times [0,\\infty)$. The boundary of $\\cal B$ parametrizes the Pappus representations and the interior consists of Anosov representations."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"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.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"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.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"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.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The Barbot component of discrete faithful representations of the modular group into Isom(SL3(R)/SO(3)) is homeomorphic to R² × [0, ∞).","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"7dd7709a4b4aad2a40c7807afc476eb9c407e5ebe95135c2369605ab7e1c7756"},"source":{"id":"2605.15317","kind":"arxiv","version":1},"verdict":{"id":"914f315b-881d-4ddf-8171-dd58324c4e67","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T15:37:11.278248Z","strongest_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.","one_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.","pipeline_version":"pith-pipeline@v0.9.0","weakest_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.","pith_extraction_headline":"The Barbot component of discrete faithful representations of the modular group into Isom(SL3(R)/SO(3)) is homeomorphic to R² × [0, ∞)."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.15317/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_title_agreement","ran_at":"2026-05-19T16:01:18.155524Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T15:54:07.324374Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T14:41:54.212116Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T13:33:22.770624Z","status":"skipped","version":"1.0.0","findings_count":0}],"snapshot_sha256":"3ffc13c4072aed9f4ee4825cf7619f0f4d59213cd60575a140c03cab656c0b4d"},"references":{"count":13,"sample":[{"doi":"","year":null,"title":"Hence tr( ρ(σ3σ2σ3σ2)) is a smooth function on the smooth part of R","work_id":"958bccd0-aafe-4ea9-a37b-8a3fdff21cac","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"τ (r2 1r2) = 64 (1 − c2)2(1 − d2)","work_id":"654307fc-b54f-4fff-b626-d31d6a562326","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"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","work_id":"f0a36590-6470-4fc5-a694-83b9c7b97cfc","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"When b ∈ [1 + √ 2, ∞ ) we have a ∈ (0, ∞ )","work_id":"887196d7-c1b3-4ca3-b268-b0dc9f16ff3d","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"See [ BL V, Eq","work_id":"f7bfccc4-2111-4d15-9d4b-92d539debe40","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":13,"snapshot_sha256":"12d45e1acd4bb3cdfc8e1b19b9e1547ceb65d24dc9912a543f86065f5fc6687c","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"7bf8c93946908ca23942216e76283cc0dcc26e209797ba2b7a4b111ff6f96a0f"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}