The Real Spectrum Compactification of Character Varieties and its relationship with other compactifications.

We study the moduli space of marked hyperbolic surfaces and related objects. Thurston shows that this space can be identified with the space of faithful and discrete representations of a finitely generated group in $\mathrm{PSL}_2(\mathbb{R})$, up to post-conjugation by $\mathrm{PSL}_2(\mathbb{R})$. This space is a key connected component of the character variety.

In this seminar we examine degenerations of these representations by studying compactifications of the character variety. In particular we present the real spectrum compactification, its topological properties, and show that it projects continuously onto the oriented compactification of the character variety, defined by Maxime Wolff. To this end, we interpret its boundary points geometrically and associate oriented real trees with them.