Groups generated by dehn twists around the curves

The mapping class group of an oriantable surface S genus g , boundary b and marked points p is the group of isotopy classes of all oriantation preserving homeomorphisms of the surface which are identity on the boundary and taking marked points to each other. One of the infinite order element of the mapping class group is Dehn twists. In this thesis we researched the groups of generated by Dehn twists about two esential simple closed curves from Ishida's article (Ishida, 1996). Then we focused on the question of groups generated by Dehn twists about three esential simple closed curves. We included recent studies on this subject.