Homotopy of curves and mappings and the Teichmüller metric (Q1360829)

Let \(T\) be the Teichmüller space consisting of equivalence classes of finite bordered Riemann surfaces of given genus and connectivity with a given finite number of functions with a topological determination (``marked''). For the model surface \((S_0,id)\) and \((S,f)\) the family \({\mathcal F}\) of homotopy classes \(\Gamma\) determined by Jordan curves on \(S_0\) not homotopic to a point determines a relationship to the corresponding \(f(\Gamma)\) on \(S\). Let \(m(\Gamma), m(f(\Gamma))\) be the appropriate modules. The author considers the quantity \[ m_\Gamma(0,x)= \underset {\Gamma\in{\mathcal F}} {\text{l.u.b.}} \frac12 \Biggl|\log \frac{m(f(\Gamma))} {m(\Gamma)} \Biggr| \] where \(0,x\) denote the elements of \(T\) determined by \((S_0,id)\) and \((S,f)\). It is not hard to see that this defines a distance from 0 to \(x\). Defining the corresponding quantity \(m(x,x')\) for \(x,x'\) determined by \((S,f)\), \((S',f')\) one gets a metric on \(T\). It is not difficult to see that this is equivalent to the Teichmüller metric.