Geometric triangulations and flips (Q2324102)

A topological triangulation of a topological surface with marked points and boundary (possibly empty) is a maximal family of topological arcs connecting marked points in such a way that they do not intersect themselves or each other in their interior. The arcs cut out the surface into ideal triangles (vertices and edges may be not distinct). A flip is a transformation of an ideal triangulation that removes an edge that is a diagonal of a quadrilateral and replaces it by the other diagonal. Here, a flat surface is a topological compact surface with an everywhere flat metric outside of a finite set of conical singularities (of arbitrary angle) and boundary formed by a finite (possibly empty) union of geodesic segments connecting conical singularities. The boundary does not need to be connected. For a given flat surface, a geometric triangulation is a topological triangulation whose edges (including the boundary) are geodesic segments and whose vertices are conical singularities (every conical singularity should be a vertex of the triangulation). The main result of this article is the following: Theorem. For a given flat surface, any pair of geodesic triangulations can be connected by a chain of flips.