{"paper":{"title":"Pseudo-Anosov flows and the geometry of Anosov-like group actions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"The action induced by a pseudo-Anosov flow on the orbit space is isometric on a Gromov-hyperbolic space.","cross_cats":["math.GT"],"primary_cat":"math.DS","authors_text":"Abdul Zalloum, Kathryn Mann, Neige Paulet, Thomas Barthelm\\'e","submitted_at":"2026-05-13T00:13:52Z","abstract_excerpt":"We show that the action on its orbit space induced by a pseudo-Anosov flow on a closed $3$-manifold (and more general Anosov-like actions) can be seen as an isometric action on a Gromov-hyperbolic space. When the flow is not $\\R$-covered, we show that this action admits elements that are weakly properly discontinuous and deduce that elements of $\\pi_1(M)$ that do \\emph{not} represent a periodic orbit of the flow are generic for any word metric coming from a finite generating set. We also give a number of other geometric group-theoretic results for Anosov-like group actions on bifoliated planes"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"the action on its orbit space induced by a pseudo-Anosov flow on a closed 3-manifold (and more general Anosov-like actions) can be seen as an isometric action on a Gromov-hyperbolic space","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The flow is pseudo-Anosov or Anosov-like on a closed 3-manifold and the orbit space admits a structure allowing the induced action to be isometric on a Gromov-hyperbolic space","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Pseudo-Anosov flows on 3-manifolds induce isometric actions on Gromov-hyperbolic spaces, with generic elements in the fundamental group for non-R-covered flows.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The action induced by a pseudo-Anosov flow on the orbit space is isometric on a Gromov-hyperbolic space.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"19aac835dde4a4cc1a39d7acd3fb3bac9a7197296fcbdb799462bebf9f7fae91"},"source":{"id":"2605.12837","kind":"arxiv","version":1},"verdict":{"id":"aece40b1-2649-4f7b-8557-a7011dd8fbba","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T18:58:26.729015Z","strongest_claim":"the action on its orbit space induced by a pseudo-Anosov flow on a closed 3-manifold (and more general Anosov-like actions) can be seen as an isometric action on a Gromov-hyperbolic space","one_line_summary":"Pseudo-Anosov flows on 3-manifolds induce isometric actions on Gromov-hyperbolic spaces, with generic elements in the fundamental group for non-R-covered flows.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The flow is pseudo-Anosov or Anosov-like on a closed 3-manifold and the orbit space admits a structure allowing the induced action to be isometric on a Gromov-hyperbolic space","pith_extraction_headline":"The action induced by a pseudo-Anosov flow on the orbit space is isometric on a Gromov-hyperbolic space."},"references":{"count":30,"sample":[{"doi":"","year":1967,"title":"D. V. Anosov and G. Sina i . Some smooth ergodic systems. With an appendix by G. A. Margulis . Uspekhi Matematicheskikh Nauk [N. S.] , 22(5(137)):107--172, 1967","work_id":"519d4082-1012-4765-81b8-43a0fb00815a","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1995,"title":"Characterization of Anosov flows in dimension 3 by their weak foliations","work_id":"b849bfa7-ae94-45d2-8340-079b827a590e","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1998,"title":"Actions de groupes sur les 1-vari\\'et\\'es non s\\'epar\\'ees et feuilletages de codimension un","work_id":"5a0923ee-b7c1-455e-842c-f6e7c2de1310","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2026,"title":"[BFM25] Thomas Barthelm´ e, Steven Frankel, and Kathryn Mann","work_id":"531064ee-eecd-4f2e-9dc8-365df504dc9e","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2017,"title":"Thomas Barthelm \\'e and S \\'e rgio R. Fenley. Counting periodic orbits of Anosov flows in free homotopy classes. Commentarii Mathematici Helvetici , 92(4):641--714, 2017","work_id":"5b13fde4-4eaf-4413-88c5-22a3eb871ead","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":30,"snapshot_sha256":"869a3c1ead59f06b6904633c60912c8f56885bfd7606e35097243ff93c908a9d","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}