# Compactifications of SL(2)-character varieties of surfaces

Thurston compactified the Teichmüller space $${\cal T}(F)$$ of a closed, oriented surface $$F$$ with a piecewise-linear sphere. Furthermore, as far as I understand, its linear pieces have natural symplectic structure. I am looking for an analogous statement for $$SL(2,\mathbb C)$$-character varieties $$X(F)$$ of surfaces $$F$$.

Q1: Is there a natural extension of $$X(F)$$ whose points have some geometric structure analogous to that of Thurtson's boundary? That may be a compactification or perhaps there is something composed of piecewise complex linear pieces which is not a compactification?

Q2: In particular, what can be said about the boundary points of the Morgan-Shalen compactification (in $$\mathbb P^S$$, where $$S$$ is the set of all closed loops of $$F$$)?

Q3: There are infinitely many finite subsets $$S'\subset S$$ whose corresponding trace functions $$\tau_s,$$ $$s\in S',$$ embed $$X(F)$$ into $$\mathbb C^{S'}$$ and, consequently, compactify $$X(F)$$ by its algebraic closure in $$\mathbb C\mathbb P^{S'}$$. These compactifications have geometric structure but are not canonical. But I expect a relation between such compactifications taken for all suitable sets $$S'$$ and the Morgan-Shalen compactification. Did someone work it out?

In my recent paper Wonderful Compactification of Character Varieties (co-authored with I. Biswas and D. Ramras) we construct a compactification of any $$G$$-character variety (including surface groups) where $$G$$ is of adjoint-type.