Geometry

The original approach to Teichmüller spaces used extensively functional analysis and was highly non constructive. Recently, Fock and Goncharov introduced in the seminal paper [50] higher Teichmüller theory as a fusion of the classical theory with representation theory by an algebraic geometry approach, opening completely new perspectives. The papers by Gekhtman, Shapiro, Vainstein [68], and by Fomin, Shapiro, D. Thurston [54] should also be seen in this context. Basic ingredients of this theory are the explicit coordinate description of the Teichmüller spaces for an open surface, which goes back to W. Thurston [108] and Penner [92], and the concept of total positivity in its modern form devised by Lusztig [84,85,83].

In [50] higher Teichmüller spaces are defined, and it is shown that they parametrise certain ``positive'' discrete, faithful representations of the fundamental group of the surface to a split real simple Lie group $G$ of higher rank. These spaces are closely related to the ones studied by Hitchin [71]. In particular, for $G=\operatorname{SL}_3(\mathbb{R})$ the corresponding higher Teichmüller space turns out [51] to coincide with the space of real projective structures on $S$ studied by Goldman and Choi [69,42]. On the other hand, for $G=\operatorname{PSL}_2(\mathbb{R})$ the classical Teichmüller theory is recovered.

In [50,47] it was shown that the ${\cal A}$ and ${\cal X}$ versions of the Teichmüller and lamination spaces can be obtained as the positive real and tropical points of certain cluster ${\cal A}$- and ${\cal X}$-varieties. These objects are closely related to cluster algebras but imply more structure.

Quantum Teichmüller spaces were constructed independently by Chekhov and Fock [41] and by Kashaev [75], see also [70],[48]. In [106] a conjecture of Verlinde is further discussed, namely that the mapping class group acts on the quantum Teichmüller space in the same way as on conformal blocks of the Liouville conformal field theory.

As a result of these developments, cluster theory becomes enriched by new examples as well as new features such as duality (see also [59]), Poisson structure and quantization (see also [67,49,59]) and relations to algebraic $K$-theory and the dilogarithm.

Other work concerning the relation between Poisson geometry and cluster structures for double cells (in groups and flag varieties) include the papers of Goodearl and Yakimov [20,110] and by Kogan and Zelevinsky [79].