Complexes of curves and Teichmüller spaces (Q1175480)

Let \(S\) be a compact surface (with boundary). \textit{W. J. Harvey} [ Ann. Math. Stud. 97, 245-251 (1981; Zbl 0461.30036)] introduced a simplicial complex \(C(S)\), whose vertices are simple loops in \(S\), and stated without proof that there is a weak homotopy equivalence between \(C(S)\) and the cuspidal boundary of the Teichmüller space \(T_ S\). The author defines a contractible subset \(X_ S\) of \(T_ S\), which is invariant under the modular group \(\hbox{Mod}_ S\), and whose quotient \(X_ S/\hbox{Mod}_ S\) is compact. He shows that the boundary of \(X_ S\) in \(T_ S\) is homotopy equivalent to the geometric realization of \(C(S)\).