ABSTRACT

REAL-ANALYTIC DIFFEOMORPHISMS OF THE CIRCLE AND MAPPING CLASS GROUPS

Yüce,İlker Savaş

M.Sc., Department of Mathematics

Supervisor: Assis. Prof. Dr. Mustafa KORKMAZ

February 2000

The subgroups of the group of orientation-preserving homeomorphisms of the circle generally do not have common algebraic properties with linear groups indicated in the theorem known as Tits Alternative, which is proved by J. Tits. This is true also for the subgroups of the group of orientation-preserving, infinitely-many times differentiable homeomorphisms of the circle. But if we increase smoothness from C^infinity to C^omega, namely if we consider the group of orientation-preserving, real-analytic diffeomorphisms of the circle, we see that an alternative, called Residual Tits Alternative, is valid. This means that the subgroups of orientation-preserving, real-analytic diffeomorphisms of the circle may have some algebraic properties common with linear

groups. Indeed, the subgoups of orientation-preserving, real-analytic diffeomorphisms of the circle have algebraic properties common with the subgroups of PSL(2,R). By the help of these common properties, it is possible to prove a result which is the triviality of homomorphisms from some of the Mapping Class

Groups to the group of real-analytic diffeomorphisms of the circle.

The aim of this work is to follow these facts given above and to make an introduction to the subject Mapping Class Groups.

Keywords: Homeomorphisms, diffeomorphisms, real-analytic diffeomorphisms of the circle, mapping class groups.