Skip to content
DraconiticResearch
Search Sources API
197,768 studies31 sourcesUpdated 38 min ago
PreprintCC0arXiv

From veering triangulations to convergence actions and back again

  • Jason Fox Manning
  • Saul Schleimer
  • Henry Segerman
Open at sourcePDFDOICite

Abstract

Suppose that M is a finite-volume cusped hyperbolic three-manifold, equipped with a veering triangulation 𝒱. We prove that the action of the fundamental group of M on the veering two-sphere is a geometrically finite convergence action. Applying a result of Yaman, we deduce that the veering two-sphere is equivariantly homeomorphic to the boundary of hyperbolic space. As an application, we obtain Cannon-Thurston maps associated to veering triangulations. If 𝒱 is layered we recover the classical Cannon-Thurston map. If it is not we obtain Cannon-Thurston maps that do not come from surface subgroups. These are the first such examples in the cusped case. Finally, we implement an algorithm to draw approximations of these Cannon-Thurston maps. This improves upon previous approximations obtained by Thurston and others.

Related

  1. Veering triangulations of once-punctured torus bundles are unique
    Saul Schleimer, Henry Segerman
    arXivPreprintCC0Full textGeometry & topologyarXiv
  2. Three-manifolds that do not admit veering triangulations
    Layne Hall
    arXivPreprintarXiv licenceFull textGeometry & topologyarXiv
  3. Drilling veering triangulations with applications to pseudo-Anosov flows
    Anna Parlak, Henry Segerman
    arXivPreprintarXiv licenceFull textDynamical systemsGeometry & topologyarXiv
  4. Recognizing convergence actions as proper isometric actions on hyperbolic spaces
    Lihuang Ding, Abdul Zalloum
    arXivPreprintarXiv licenceFull textAlgebra & number theoryGeometry & topologyarXiv
  5. Convergence of spectral smoothings from compact quantum group actions
    Trevor Camper, Dimitrios Giannakis, Gage Hoefer
    arXivPreprintCC BY 4.0Full textAnalysis & PDEsarXiv

Cite

BibTeXRIS
BibTeX
@misc{manning2026from,
  title = {{From veering triangulations to convergence actions and back again}},
  author = {Jason Fox Manning and Saul Schleimer and Henry Segerman},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.14598},
  eprint = {2609.14598},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.14598},
  note = {CC0},
}
RIS
TY  - UNPB
TI  - From veering triangulations to convergence actions and back again
AU  - Jason Fox Manning
AU  - Saul Schleimer
AU  - Henry Segerman
PY  - 2026
DA  - 2026/09/13
PB  - arXiv
DO  - 10.48550/arxiv.2609.14598
UR  - https://arxiv.org/abs/2609.14598
AB  - Suppose that M is a finite-volume cusped hyperbolic three-manifold, equipped with a veering triangulation 𝒱. We prove that the action of the fundamental group of M on the veering two-sphere is a geometrically finite convergence action. Applying a result of Yaman, we deduce that the veering two-sphere is equivariantly homeomorphic to the boundary of hyperbolic space. As an application, we obtain Cannon-Thurston maps associated to veering triangulations. If 𝒱 is layered we recover the classical Cannon-Thurston map. If it is not we obtain Cannon-Thurston maps that do not come from surface subgroups. These are the first such examples in the cusped case. Finally, we implement an algorithm to draw approximations of these Cannon-Thurston maps. This improves upon previous approximations obtained by Thurston and others.
ER  -