Topological components of surface group representations and signature

We study the topological components of the surface group representations into $\mathrm{SL}(2,\mathbb{R})$ and $\mathrm{PSL}(2,\mathbb{R})$. Utilizing the signature formula established in [14], we determine the number of connected components of the representation spaces with boundary elliptic, hyperbolic, and parabolic holonomies.