Sorry in advance for my English.
I'm studying the Thurston compactification from the Jean-Pierre Otal's book "The Hyperbolization Theorem for Fibered 3-Manifolds".
I have a question, what $\mathbb{R}_{+}^\mathcal{C}$ is? I know $\mathcal{C}$ is the set of conjugacy classes, but this space is not just $\mathbb{R}_{+}$? If you can give an explicit example it would be nice! Or perhaps a drawing.
(The space $\mathbb{R}_{+}^\mathcal{C}$ is also mentioned in wikipedia:Thurston boundary)
Also if a can get a intuitive answer about the Thurston compactification it would be the best. I read the compactification theorem, but I need an example (I want to know how this compactification looks like).
Thank you all.
I'll restrict my answer to your question about $\mathbb R_+^{\mathcal C}$.
In set theory, given two sets $A,B$, the set $A^B$ denotes the set of all functions $f : B \to A$. For example, $\mathbb R_+^n$ really means $\mathbb R_+^{\{1,...,n\}}$ which means all functions $\{1,...,n\} \mapsto \mathbb R_+$ which means all sequences of positive real numbers $(x_1,...,x_n)$. You can visualize $\mathbb R_+^2$ as the open positive quadrant of the Cartesian plane, and $\mathbb R_+^3$ as the open positive octant of Cartesian 3-space. In general you can think $\mathbb R_+^n$ as the open positive orthant of Cartesian $n$-space, although I'm not sure how to advise you on "drawing" the high dimensional cases.
So $\mathbb R_+^{\mathcal C}$ is the set of functions $\mathcal C \to \mathbb R_+$. The set $\mathcal C$ is presumably countably infinite (you don't say which group, but as I recall Otal's book uses $\mathcal C$ to denote the set of conjugacy classes of the fundamental group of a finite type surface, in which case $\mathcal C$ is indeed countably infinite). So, if you were to choose an enumeration $\mathcal C = \{c_1,c_2,c_3,...\}$ then $\mathbb R^{\mathcal C}$ is an infinite dimensional version of Euclidean space, and $\mathbb R_+^{\mathcal C}$ is its positive orthant, which is the set of infinite tupes of positive numbers $(x_1,x_2,x_3,...)$. Which, I'm sorry to say, I am entirely unable to draw for you.
Your closing question about the Thurston compactification is a gigantic question, and my best advice is that you save questions about that for later, after you've dug into the proof of Thurston's compactification theorem and have developed specific mathematical questions about it. Generally speaking, asking one question at a time is the best use of this site.