On the inclusion of the quasiconformal Teichm\"uller space into the length-spectrum Teichm\"uller space

Given a surface of infinite topological type, there are several Teichm\"uller spaces associated with it, depending on the basepoint and on the point of view that one uses to compare different complex structures. This paper is about the comparison between the quasiconformal Teichm\"uller space and the length-spectrum Teichm\"uller space. We work under this hypothesis that the basepoint is upper-bounded and admits short interior curves. There is a natural inclusion of the quasiconformal space in the length-spectrum space. We prove that, under the above hypothesis, the image of this inclusion is nowhere dense in the length-spectrum space. As a corollary we find an explicit description of the length-spectrum Teichm\"uller space in terms of Fenchel-Nielsen coordinates and we prove that the length-spectrum Teichm\"uller space is path-connected.