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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.","authors_text":"Richard Evan Schwartz","cross_cats":[],"headline":"The Barbot component of discrete faithful representations of the modular group into Isom(SL3(R)/SO(3)) is homeomorphic to R² × [0, ∞).","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GT","submitted_at":"2026-05-14T18:31:01Z","title":"On Pappus and Anosov Representations of the Modular Group"},"references":{"count":13,"internal_anchors":0,"resolved_work":13,"sample":[{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":1,"title":"Hence tr( ρ(σ3σ2σ3σ2)) is a smooth function on the smooth part of R","work_id":"958bccd0-aafe-4ea9-a37b-8a3fdff21cac","year":null},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":2,"title":"τ (r2 1r2) = 64 (1 − c2)2(1 − d2)","work_id":"654307fc-b54f-4fff-b626-d31d6a562326","year":null},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":3,"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","year":null},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":4,"title":"When b ∈ [1 + √ 2, ∞ ) we have a ∈ (0, ∞ )","work_id":"887196d7-c1b3-4ca3-b268-b0dc9f16ff3d","year":null},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":5,"title":"See [ BL V, Eq","work_id":"f7bfccc4-2111-4d15-9d4b-92d539debe40","year":null}],"snapshot_sha256":"12d45e1acd4bb3cdfc8e1b19b9e1547ceb65d24dc9912a543f86065f5fc6687c"},"source":{"id":"2605.15317","kind":"arxiv","version":1},"verdict":{"created_at":"2026-05-19T15:37:11.278248Z","id":"914f315b-881d-4ddf-8171-dd58324c4e67","model_set":{"reader":"grok-4.3"},"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","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, ∞).","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.","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."}},"verdict_id":"914f315b-881d-4ddf-8171-dd58324c4e67"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:e79d7c63ca1b8b3e5cf7ec3a91491eb768557a1146cce6acf644ecb8d102a636","target":"record","created_at":"2026-05-20T00:00:52Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"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}},"canonical_sha256":"e62311568cce5a6b6dc5d4584c6c5b860c6a77d58bac98543b12c8baf25cd002","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e62311568cce5a6b6dc5d4584c6c5b860c6a77d58bac98543b12c8baf25cd002","first_computed_at":"2026-05-20T00:00:52.348685Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-20T00:00:52.348685Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"CSzLM4gV7w5DiMyoij4j/NlEyN9AcdIhGwEvCflZ4EGCMGF5jUO11QX95CNGRG3CBvvCW375RJEux7Y0Dfk6BQ==","signature_status":"signed_v1","signed_at":"2026-05-20T00:00:52.349289Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.15317","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e79d7c63ca1b8b3e5cf7ec3a91491eb768557a1146cce6acf644ecb8d102a636","sha256:3fc6be7d60b2628d8018f9ce41fbd9ce1b9878a41293642fbabb3df1d11dff51"],"state_sha256":"134b7c2a45a1cd535da1fff4240a7630106ecf11cf7abcd47eccb05f1279e89f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2J7aCpeaZvQkZNxnan3mVT/9uMLeRLkmq3IRH2CbFAd0QnGbCeCtRr1tkg11VGUSgt10X+wr3xWGK2v76U8xDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T13:18:55.423234Z","bundle_sha256":"3ca82c94b0605b6443c933fbe985d29f2afbf351f4dcb228b9b6f20df74a067b"}}