Broken hyperbolic structures and affine foliations on surfaces.
Let \(S\) be an oriented surface of negative Euler characteristic with at least one puncture. The authors investigate the following extension of Teichmüller space for \(S\): First, choose an ideal triangulation \( \lambda\) of \(S\), i.e., a decomposition of \(S\) into triangles whose edges are homotopically nontrivial arcs with endpoints at the punctures of \(S\). It is well known that Teichmüller space can be described by glueing coordinates where ideal hyperbolic triangles are glued isometrically along their boundaries according to the combinatorial pattern described by \(\lambda\) and with the additional condition that the glueing weights along all sides going into a fixed puncture add up to zero. Here, the authors allow that the glueing is by homotheties of the edges which are not isometries, with a condition for each cusp corresponding to the weight condition at the punctures for Teichmüller space. In this case, each weight system describing the homotheties defines an element \(\varphi \in H^1 (S; \mathbb R)\). If this element is fixed, then the corresponding system of structures equipped with a suitable topology is shown to be homeomorphic to a ball of dimension \(6g - 6 + 2c\) where \(c > 0\) is the number of punctures. Moreover, it can be compactified by adding a boundary sphere which in turn can be described as a family of ``broken measured foliations'', a generalization of the usual space of measured foliations which compactifies Teichmüller space.
Teichmüller space for surfaces with punctures
scientific article; zbMATH DE number 2124643