SMS scnews item created by Catherine Meister at Mon 18 Aug 2025 0914
Type: Seminar
Distribution: World
Expiry: 8 Sep 2025
Calendar1: 20 Aug 2025 1300
CalLoc1: SMRI Seminar Room (Macleay Building A12 Room 301)
CalTitle1: Geometry and Topology Seminar
Auth: cmeister@staff-10-48-22-201.vpnuser.sydney.edu.au (cmei0631) in SMS-SAML
Geometry and Topology Seminar: Tang
Straightening structures on surfaces
Robert Tang, Xi’an Jiaotong-Liverpool University
Geometry and Topology Seminar, 20th August 2025
1 pm – 2 pm, SMRI Seminar Room (A12 Macleay Room 301)
Abstract: Triangulations are ubiquitous throughout low-dimensional topology and geometry. It is a well-known result that any two triangulations of a topological surface are related by a finite sequence of flips. The analogous result is also known to be true for geometric triangulations for surfaces equipped with certain geometric structures, for example, Euclidean cone metrics or hyperbolic surfaces with a fixed vertex set. However, the prevailing methods for studying triangulations and flip sequences, in the geometric setting, tend to be very specific to the type of geometry involved.
In this talk, I will introduce straightening structures on surfaces. This provides an axiomatic framework which models the behaviour of geodesic paths on non-positively curved surfaces. Moreover, it permits a notion of 'straight triangulation', generalising the geometric triangulations from the Euclidean or hyperbolic settings. Our main result is that the 'straight flip graph' associated to a straightening structure is non-empty, connected, and quasi-isometrically embedded in the flip graph of the underlying topological surface. I will also describe other classes of geometric structures that give rise to straightening structures.
This is joint work with Valentina Disarlo.
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