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Discrete Systolic Inequalities
Kowalick, Ryan

2013, Doctor of Philosophy, Ohio State University, Mathematics.
Gromov’s systolic estimate, first proved in [2], is considered one of the deepest
results in systolic geometry. It states that, for an essential Riemannian n-manifold M,
the length of the shortest noncontractible loop, or systole, of M, denoted Sysp1 (M)

The formula can be viewed in the abstract of the actual dissertation on Ohio Link.

where the constant Cn only depends on n and not on M.
We will prove a discrete version of related theorems for triangulated surfaces.
The argument involves creating a special Riemannian metric on a triangulated surface
whose total volume is close to the number of facets in the triangulation. This
metric then allows one to convert Riemannian geodesics to homotopic edge paths of
controlled length. The proof of the analogous inequality in the case of a triangulated
triangles then follows easily from these facts.
We then apply our discrete version to facts about triangulations of orientable
surfaces. Given a triangulated, orientable, closed surface with x 2-simplices, we can
ask how many 3-simplices are required to “fill” the triangulation: that is, produce
a triangulated 3-manifold whose boundary triangulation is the triangulated surface
with which we started. Our method produces such a 3-manifold with no more than
O(x log2 x) simplices.
We will also prove that a discrete version of this inequality implies the Riemannian
version. The proof of this fact involves creating a triangulation of a Riemannian
manifold that is in some sense aware of the geometry of the manifold. We embed M
in Rm using the Nash Embedding Theorem [5] and use an argument of Whitney’s [6]
to produce a triangulation whose simplices are large in volume relative to their edges
in the metric of Rm. By working on a small enough scale, one obtains information
about the geometry of the simplices embedded in M in the induced path metric.
Since M is isometrically embedded, this gives the result. Again, once this obtained,
proving that a discrete systolic inequality implies a Riemannian one is quite simple.
Finally, we present Whitney’s argument and the necessary modifications needed
to obtain our triangulation theorem for embedded submanifol
Jean-Francios Lafont (Advisor)
Michael Davis (Committee Member)
Matthew Kahle (Committee Member)
61 p.

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