Connected components of the compactification of representation spaces of surface groups (Q633898)

Let \(\Gamma\) be a discrete group with a given finite generating set \(S\). Let \(R_{\Gamma}(n) =\text{ Hom}(\Gamma, \text{Isom}^{+}(H^{n}))\). Denote by \(m_{\Gamma}^{u}(n)\) the biggest Hausdorff quotient of \(R_{\Gamma}(u)/\text{Isom} (H^{n})\). Let \(\overline{m_{\Gamma}^{u}(n)}\) be the closure of \(m_{\Gamma}^{u}(n)\) in the space \(m_{\Gamma}^{u}(n)\cup {\mathcal{T}}(\Gamma)\) with the modified equivariant Gromov topology, where \({\mathcal{T}}(\Gamma)\) is the set of actions on lines and actions on \({\mathbb{R}}\)-trees with certain conditions. The author focuses his interest on the case where \(\Gamma\) is the fundamental group \(\pi_{1}\Sigma_{g}\) of a closed, oriented, connected surface \(\Sigma_{g}\) of genus \(g\geq 2,\) with a given standard presentation. In this case \(m_{\pi_{1} \Sigma_{g}}^{u}(2)\) is abbreviated to \(m_{g}^{u}\). Let \(m_{g,k}^{u}\) denote the connected component of \(m_{g}^{u}\) consisting of classes of representations whose Euler class, in absolute value, equals \(k\). The main result in this paper is: Theorem 1.1 Let \(g \geq 4\) and \(k \in \{0, ..., 2g-3\}\). Then, in \({\overline {m_{g}^{u}}}\), the boundary of the Teichmüller\enskip space, \(\enskip \partial m_{g,2g-2}^{u}\), is contained in \(\partial m_{g,k}^{u}\) as a closed, nowhere dense subset. This theorem implies that the compactification \({\overline {m_{g}^{u}}}\) is extremely wild. This is a strong contrast to the Thurston compactification. To prove this theorem, the author uses the fact that the connected components of \(m_{g}^{u}\) are one-ended.