| This talk will be on an application of Klein-Maskit Combination Theorems in the construction of a Riemann surface out of Y-pieces and Q-pieces. We would like to represent a Rieman surface as X=D/G, where D is the Poincare disk and G is a Fuchsian group acting in D. In order to do that we start with some groups of 2 generators and then try to glue the pieces corresponding to D mod the action of those groups. By gluing two pieces we mean that the group generated by 4 generators has to be discontinuous and secondly it has to uniformize X. We introduce Klein-Maskit combination theorem as a result of which the above can be achieved. |