Y-Pieces and Twist Parameters

Abstract

In this chapter we carry out the construction of the compact Riemann surfaces and introduce the Fenchel-Nielsen parameters as outline in Section 1.7. The building blocks are the pairs of pants or Y-pieces, whose geometry will be studied in some detail in Section 3.1. We then give a first idea of Teichmüller space by considering the so-called marked Y-pieces. If two Y-pieces are pasted together, one obtains an X-piece, and an additional degree of freedom apprears: the twist parameter. We show that the twist parameter can be computed in terms of the lengths of closed geodesics and vice-versa. In Section 3.5 we briefly consider cubic graphs which are the underlying combinatorial structure in the construction of the surfaces. We also give an estimate of the number of such possible graphs. The construction then follows in Sectuib 3.6 at the end of which we give a large number of pairwise non-isometric examples. The final section is an essay on the multiplicity of the length spectrum and is considered as a first application. This section is an appendix and may be skipped in a first reading.