Thurston compactified the Teichmuller 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 $F$)?

Related 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?

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?

Thurston compactified the Teichmuller 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.

Q1 Is there a natural compactification of $SL(2,\mathbb C)$-character varieties $X(F)$ of surfaces whose points at infinity have some geometric structure (perhaps analogous to that of Thurtson's boundary)?

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?

Thurston compactified the Teichmuller 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 $F$)?

Related 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?