Quote by tom.stoer
I think my example with Sł was missleading; I'll try it again:
The dimension of the vector space of Cfunctions on Sł is infinitedimensional. But if we look at the Lie group SU(2) we can write its elements as
[tex]U[f] = \exp\left\{i t^a f^a(x)\right\}[/tex]
The functions f^{a} are members of an infinite dimensional vector space, but nevertheless we talk about a finite dimensional group b/c we have finitly many generators t^{a}.

Diffeomorphisms which are infinitesimally close to the identity look like gauge transformations. The problem is that infinitesimal diffeomorphisms do not generate the whole group.
Looking at the Ashtekar formulation of gravity there are the Gausslaw generators G^{a}(x) and the spacelike diffeomorphism generators = vectorconstraints V^{a}(x). The transformations are generated via functionals like
[tex]G[f] = \int d^3x\,G^a(x)\,f(x);\quad V[f] = \int d^3x\,V^a(x)\,f(x);\quad[/tex]
Again we have finitly many generators even if the functions f belong to an infinite dimensional vector space.

If there are a finite number of generators, we are not talking about the full diffeomorphism group, only the infinitesimal diffeomorphisms.