Thurston's metric on Teichmüller space of semi-translation surfaces

We define two pseudo-metrics L_F and K_F on the Teichmüller space of semi-translation surfaces, which are the symmetric counterparts to the metrics defined by William Thurston. We prove some nice properties of L_F and K_F, most notably that they are complete pseudo-metrics. In the second part of the thesis we define their asymmetric analogues L_F^a and K_F^a and prove that their equality depends on two statements regarding 1-Lipschitz maps between polygons. We are able to prove the first statement, but the second one remains a conjecture: nonetheless, we explain why we believe it is true.