|M.Sc Student||Shir Sivroni|
|Subject||Free Actions of Surface Groups on R-Trees|
|Department||Department of Mathematics||Supervisor||Full Professor Moriyah Yoav|
In this work, we proved, that given a compact hyperbolic surface without boundary F, There exists an R-tree, T, on which the fundamental group of F, is acting by isometries. We refered, in this work, only to compact hyperbolic surfaces without boundary. We also refered to the covering space of F, as the disc model for the hyperbolic plane. A geodesic in F, is the image of a geodesic in the hyperbolic plane, under the covering map.In order to build the R-tree, T, we used, the theory of laminations and of measured laminations. In particular, we used stable laminations of a certain kind of automorphisms of F, which are called irreducible and non periodic. A geodesic lamination L, on a surface F, is a closed subset of F, which is a disjoint union of geodesics. A lamination L’ in the hyperbolic plane, is a disjoint union if geodesics in the hyperbolic plane, which is equivariant under the action of the fundamental group of F, on the hyperbolic plane. Let L be a lamination in a surface F, which is the stable lamination of an automorphism h of F, which is irreducible and non-periodic. Then, The preimage of L in the hyperbolic plane, under the covering map p, is a lamination L’, in the hyperbolic plane, with certain properties. Yhe geodesics of L’, are either boundary geodesics of av ideal finite polygon in L’, and therefore, are isolated in L’ from one side, or geodesics which are not isolated from any side. The points of T, are defined to be closed ideal polygons in the hyperbolic plane, with boundary, which is a union of geodesics from L’, or geodesics from L’, which are not isolated. We use the theory of measured lamionations, in order to define the metric on T, and we use the action of the fundamental group of F on the hyperbolic plane, to show that the fundamental group of F, is acting on T, by isometries.