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?


I don't think this actually answers any of your questions explicitly, but it might help.

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.

In the free group case (open surface group), we also show the compactification is simply-connected, admits a natural Poisson structure, and identify the boundaries as "parabolic character varieties".

For the closed surface group case you seem to be interested in, you can try to "cut-out" the surface group boundaries from the free group case.

Regardless, the introduction of the above paper has many references that might be useful to you (including work by Manon, and also Parreau). In particular, the work of Manon shows that for each quiver-theoretic avatar of a free group character variety discussed here there is a natural compactification (this might speak to your Question 3). The work of Parreau might speak to your interest in geometrically significant boundary divisors.

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