On the combinatorics of veering triangulations

Lead Research Organisation: University of Warwick
Department Name: Mathematics

Abstract

In the 1970's and 1980's William Thurston revolutionised low-dimensional topology. He introduced many new ideas into the field, and furthermore revealed exciting connections between the combinatorial study of three-manifolds and many other areas:
Teichmuller theory, holomorphic dynamics, Kleinian groups, geometric group theory, and more.

One of the new techniques he pioneered was the use of ideal triangulations to represent hyperbolic three-manifolds. This allowed topologists, on the one hand, access to tools in algebraic geometry and, on the other hand, an organising principle for what had been a hodge-podge of disparate examples. The idea of ideal triangulations has now been generalised and specialised many times to give angled triangulations [Casson], taut ideal triangulations [Lackenby], and veering triangulations [Agol, HRST].

Francois Gueritaud has found an important connection between the Cannon-Thurston map for fibered manifolds and their veering triangulations. Yair Minsky and Samuel Taylor have given an interesting relation between the machinary of subsurface projections and the structure of veering triangulations. The goal of this project is to understand how the combinatorics of a veering triangulation, in the non-fibered setting, inform the cooresponding Cannon-Thurston map and the "fibered structure" of the universal cover of the three-manifold.

Publications

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Studentship Projects

Project Reference Relationship Related To Start End Student Name
EP/N509796/1 01/10/2016 30/09/2021
1936817 Studentship EP/N509796/1 02/10/2017 31/05/2021 Anna Barbara Parlak
 
Description Oberwolfach Leibniz Graduate Students
Amount € 200 (EUR)
Organisation Mathematical Research Institute of Oberwolfach 
Sector Academic/University
Country Germany
Start 02/2020 
End 02/2020
 
Description Research visit, host: Richard Kenyon
Amount $100 (USD)
Organisation Yale University 
Sector Academic/University
Country United States
Start 07/2019 
End 07/2019
 
Description Research visit, host: Samuel Taylor
Amount $83 (USD)
Organisation Temple University 
Sector Academic/University
Country United States
Start 07/2019 
End 07/2019