Geodesic currents and Teichmüller space (Q1765327)

In this paper, the author defines a Thurston-type boundary for the Teichmüller space \({\mathcal T}(X)\) of a surface \(X\) of infinite area by using the embedding of Teichmüller space into the space of Liouville currents. The theory of Liouville currents in such a setting was developed by F. Bonahon for the case of surfaces of finite area. The basic tool that is used in the paper under review is the Liouville map \({\mathcal L}\) that sends \({\mathcal T}(X)\) into the space \({\mathcal H}(X)\) of Liouville currents on \(X\). The author shows that the map \({\mathcal L}\) is a homeomorphism onto its image and that the image \({\mathcal L}({\mathcal T}(X))\) is closed and unbounded in \({\mathcal H}(X)\). Since this image is closed, it has no new ``boundary points'', but since it is unbounded, one can define a natural boundary to \({\mathcal T}(X)\) by adding to it the set of rays asymptotic to \({\mathcal L}({\mathcal T}(X))\). The author then identifies these asymptotic rays with bounded measured laminations in \(X\). In that way, the Thurston-type boundary for \({\mathcal T}(X)\) becomes the space of projective bounded measured laminations on \(X\). The closure of the Teichmüller space \({\mathcal T}(X)\) is not compact (in this setting of infinite area surfaces), but there is still a natural action of the quasiconformal mapping class group of the surface as a group of homeomorphisms of the union of \({\mathcal T}(X)\) with its Thurston-type boundary. The paper contains also results on earthquake paths and on their limit points on the boundary.