Or: some rambling thoughts on the following questions—in what sense is Outer space “like Teichmüller space for “? And in what sense is “higher Teichmüller theory” Teichmüller theory?
The classical / canonical case
For a closed surface of genus , Teich(S) is the set of hyperbolic metrics on S, modulo isotopy. (By the uniformization theorem, these are all the constant-curvature metrics on S.) We may specify points in Teich(S) by pairs (X, h) where X is a surface of genus g (the “marked surface”, or “clothing”) with a hyperbolic metric, and (the “marking”, or “instructions on how to wear the clothing”) is a homeomorphism. Alternatively, we may specify points by specifying a (hyperbolic) Riemannian metric m. We may go between the two notions: given a pair (X, h), the metric m in question is the pullback of the metric on X under h.
The construction of the Fenchel-Nielsen coordinates shows that Teich(S) is topologically an open (6g-6)-cell. We can endow it with a topology in various ways (algebraically, etc.)
Teich(S) can be given a Finsler metric, the Teichmüller metric , with respect to which it is complete but not nonpositively-curved; it can also be given a Riemannian metric, the Weil-Petersson metric (which has many reinterpretations—see e.g. Thurston-Wolpert, Bonahon, McMullen), with respect to which it is negatively-curved but not complete. The metric completion with respect to is precisely the compactification of Teich(S) given by augmented Teichmueller space.
The mapping class group Mod(S) acts naturally on Teich(S), by precomposition of markings. This can be used to do things such as classify elements of Mod(S) (Thurston), study deformations of Klenian groups (Minsky-Canary-many others), etc.
“Slightly elevated Teichmueller theory”
Choi-Goldman constructed the set C(S) of convex real projective geometric structures on a closed surface S [modulo some natural notion of equivalence]. They give it a topology analogous to the algebraic topology on Teich(S), and aso show that it is a (16g-16)-cell, using coordinates analogous to the Fenchel-Nielsen coordinates.
Moreover, this space has the natural structure of a complex variety (see e.g. this Calegari blogpost for an exposition.)
Other things called higher Teichmüller theory
… or that have that label attached to them: the Hitchin Component—but this seems like a very much more algebraic generalisation of Teich(S) (whereas Teich(S) may be seen as a connected component of a space of representations from to PSL(2,R), Hitchin components are connected components of spaces of representations into PSL(n,R) for more general n.
Ditto more general spaces of representations [albeit representations with nice geometric properties / motivations, e.g. Anosov representations]
Fock-Goncharov have work in a similar vein (representations of into a more general semisimple Lie group.) [see reference below]
The case of outer space
Outer space may be defined as the set of all metric tree structures on a bouquet of n circles (slightly more poetically, a rose of n petals), modulo free homotopy. Points in outer space may be specified as pairs (X, h) where X is a graph homotopy-equivalent to the bouquet of n circles, and h is a free homotopy class of homotopy equivalences from X to said bouquet.
It has the structure of a simplicial structure (with open simplices—most vertices are missing.)
, which may be described as the mapping class group of a bouquet of n circles, acts naturally on by precomposition of markings.
The common theme
Given a topological space X, “the Teichmüller space of X” is (should be?), roughly speaking, the set of all “natural [geo]metric structures” on X; a priori this may only be a set, but it may turn out that it can be given additional structure, possibly in nice ways.
I’m not sure there is a systematic way to make “natural [geo]metric structure” more precise at the moment, or that we can more systematically specify when and how we can endow what is a priori the set that is Teichmüller space with more structure, or at least not ways of doing this which encompass all the contexts described above.
There is a paper of Grothendieck which axiomatizes (classical) Teichmüller space using the machinery of analytic (and algebraic) geometry, obtaining a description in terms of principal fibre bundles over complex varieties / algebraic curves, but this does not seem to extend to the other contexts (at least not in a straightforward / natural way.
This level of (im)precision is not unlike that which surrounds the notion of a moduli space at the moment. (See Wikipedia article on moduli spaces: “Moduli spaces are spaces of solutions of geometric classification problems”)
Maybe it’s more productive that way, at least as things stand—to leverage the intuitive power of the metaphor without explicitly worrying about trying to make the metaphor categorically precise.