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Published on January 3rd, 2014 | by Emily Corbett

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# Cross Ratios and Thurston’s Gluing Equations over Rings

By Montek Gill, University of Sydney

In the world of mathematics, gluing is a precise, mathematical process of constructing new spaces from given ones. It is really just the translation of the usual, everyday idea of gluing into the language of mathematics. For example, the first example many people see of mathematical gluing is the construction of a torus from a square, as shown below.

We may apply our gluing procedure to many different types of spaces, but not all gluings will result in “nice” spaces, or what mathematicians call, manifolds. In three dimensions, it is common to glue tetrahedra together to form new spaces. William Thurston (1946 – 2012) developed a certain set of equations, over the complex numbers, now known as the hyperbolic gluing equations, or Thurston’s gluing equations, which described when hyperbolic tetrahedra could be glued together to form a hyperbolic manifold. One could attempt to solve these equations to find gluings which result in hyperbolic manifolds.

However, Feng Luo generalised this situation. The gluing equations of Thurston involve only integer coefficients and so the equations may be studied over general rings (with identity) rather than just the complex numbers. Luo considered general, not necessarily hyperbolic, three dimensional manifolds and their triangulations (that is, sets of tetrahedra along with gluing procedures which glue these tetrahedra to produce the manifolds), and then defined Thurston’s equations for these gluings over general rings. In his paper “Solving Thurston Equation in a Commutative Ring” Luo proved a theorem which showed that, given that a solution to the gluing equations exists over some commutative ring, then important information may be ascertained about the original manifold.

This then motivates the study of the existence of solutions in rings to Thurston’s gluing equations for triangulations of three dimensional manifolds. In Luo’s aforementioned paper, he worked out simple examples by hand using small rings which showed whether or not solutions exist in those rings. Our direction of research is to use the computer software Regina and Singular to first provide us with triangulations of well-known, three dimensional manifolds and then to study solution sets, over various rings, of the gluing equations corresponding to these triangulations. The aim is to develop general procedures, for particular rings, for determining whether or not a solution exists to the gluing equations.

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