On the compactification of Teichmüller-like parameter spaces

The construction of compactifications of Teichmüller spaces in the approach of Morgan and Shalen has close relationships with tropical geometry. By studying the general properties of the logarithmic limit set of real semi-algebraic sets, it is possible to generalize their construction and to understand some of its properties. When applied to Teichmüller spaces, this gives the compactification of Thurston, and the natural piecewise linear structure of the boundary appears automatically, showing clearly how this structure is related with the semi-algebraic structure of the interior part. It is also possible to construct a compactification of the parameter spaces of strictly convex projective structures on a closed n-manifold. In this case objects from tropical geometry also appear in the interpretation of the boundary points.