another interpretation of Teichmüller space is as a representation space for surface groups. If is hyperbolic, of finite type and is the fundamental group of then Teichmüller space is in natural bijection with: The set of injective representations with discrete image, up to conjugation by an element of , if is compact ; In general, the set of such representations, with the added condition that those elements of which are represented by curves freely homotopic to a puncture are sent to parabolic elements of , again up to conjugation by an element of . The map sends a marked hyperbolic structure to the composition where is the monodromy of the hyperbolic structure and is the isomorphism induced by . Note that this realises as a closed subset of which endows it with a topology. This can be used to see the homeomorphism directly. This interpretation of Teichmüller space is generalised by higher Teichmüller theory, where the group is replaced by an arbitrary semisimple Lie group. A remark on categories All definitions above can be made in the topological category instead of the category of differentiable manifolds, and this does not change the objects. Infinite-dimensional Teichmüller spaces Surfaces which are not of finite type also admit hyperbolic structures, which can be parametrised by infinite-dimensional spaces (homeomorphic to ). Another example of infinite-dimensional space related to Teichmüller theory is the Teichmüller space of a lamination by surfaces. Action of the mapping class group and relation to moduli space The map to moduli space There is a map from Teichmüller space to the moduli space of Riemann surfaces diffeomorphic to , defined by . It is a covering map, and since is simply connected it is the orbifold universal cover for the moduli space. Action of the mapping class group The mapping class group of is the coset group of the diffeomorphism group of by the normal subgroup of those that are isotopic to the identity (the same definition can be made with homeomorphisms instead of diffeomorphisms and, for surfaces, this does not change the resulting group). The group of diffeomorphisms acts naturally on Teichmüller space by If is a mapping class and two diffeomorphisms representing it then they are isotopic. Thus the classes of and are the same in Teichmüller space, and the action above factorises through the mapping class group. The action of the mapping class group on the Teichmüller space