Triangulated surfaces and polyhedral structures

In this chapter we introduce the foundational material that will be used in our
analysis of triangulated surfaces and of their quantum geometry. We start by
recalling the relevant definitions from Piecewise–Linear (PL) geometry, (for which
we refer freely to [20, 21]). After these introductory remarks we specialize to the
case of Euclidean polyhedral surfaces whose geometrical and physical properties
will be the subject of the first part of the book. The focus here is on results which
are either new or not readily accessible in the standard repertoire. In particular we
discuss from an original perspective the structure of the space of all polyhedral
surfaces of a given genus and their stable degenerations. This is a rather delicate
point which appears in many guises in quantum gravity [6], and string theory, and
which is related to the role that Riemann moduli space plays in these theories.
Not surprisingly, the Witten–Kontsevich model [10] lurks in the background of our
analysis, and some of the notions we introduce may well serve for illustrating, from
a more elementary point of view, the often deceptive and very technical definitions
that characterize this subject. In such a framework, and in the whole landscaping
of the space of polyhedral surfaces an important role is played by the conical
singularities associated with the Euclidean triangulation of a surface.We provide, in
the final part of the chapter, a detailed analysis of the geometry of these singularities.