A closed ball compactification of a maximal component via cores of trees

We show that, in the character variety of surface group representations into the Lie group PSL.2; R/ PSL.2; R/, the compactification of the maximal component introduced by the second author is a closed ball upon which the mapping class group acts.We study the dynamics of this action.Finally, we describe the boundary points geometrically as .A 1 A 1 ; 2/-valued mixed structures.53C43, 57K20 1. Introduction 3693 2. Background 3697 3. Core of a product of trees 3699 4. Thurston's compactification 3703 5. Fixed point for the mapping class group action 3706 6. a C -valued measured laminations and mixed structures 3710