The Teichmüller space of compact Riemann surfaces of genus \(g\) is a ball of dimension \((6g-6)\). In this paper an extension of the ordinary Teichmüller space is studied. Every conformal structure on a compact surface \(\Sigma\) can be uniformized as the quotient of the upper half plane by a subgroup of \(PSL(2,\mathbb{R})\) isomorphic to the fundamental group \(\pi_ 1(\Sigma)\). This describes a homomorphism from \(\pi_ 1(\Sigma)\) to \(PSL(2,\mathbb{R})\) well defined up to conjugation. The set of all such homomorphisms constitute a connected component of the topological space \(\Hom(\pi_ 1(\Sigma);PSL(2,\mathbb{R}))/PSL(2,\mathbb{R})\). Hence one of the components of this space must be homeomorphic to a Euclidean space \(\mathbb{R}^{6g- 6}\). In this paper the group \(PSL(2,\mathbb{R})\) is replaced by \(PSL(n,\mathbb{R})\) or more generally by the adjoint group of a split real form \(G^ r\) of any complex simple Lie group \(G^ c\). It is shown that there exists an analogous component homeomorphic to \(\mathbb{R}^{(2g-2)\dim G^ r}\) which contains ordinary Teichmüller space in a rather canonical way. The author also studies the other components for the case \(PSL(n,\mathbb{R})\). The results are proved using the theory of Higgs bundles.