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Haagerup z˜ η w ˜ x ξ y z η w x x ξ =u y n,l y ξ,η y w u ξ,η m,r . Note that the vertices e, f and g are fixed by taking ∼. We easily know that z w u n,l t z˜ w =u n,l =u z w n,r . Note that the multiplicity n of the edges z-w is equal to that of the edges z-w. ˜ Now we begin to construct a candidate for the list of gauge transformation matrices. First, for the connections M (∗/6) and M (˜∗/6), we fix the gauges for the simple edges as u ∗ G a 6 =u 1,l ∗˜ G =u 1,r a˜ 6 =u 1,l = (1)1,1 .

Of Tokyo Seminary Note 45, recorded by Y. : Paths on Coxeter Diagrams: From Platonic solids and singularities to minimal models and subfactors. : Classification of subfactors: Reduction to commuting squares. Invent. Math. : Classification of amenable subfactors of Type II. Acta Math. : Two subfactors arising from a non-degenerate commuting squair II – tensor categories and TQFT’s. Internat. J. Math. : Commuting squares and index for subfactors. D. S. : On the non-occurrence of the Coxeter graphs β2n+1 , E7 , D2n+1 as principal graphs of an inclusion of II1 factors.

There exists a family of maps: θ → πθ : C ∞ (T2 ) → T2θ such that for each θ the range of πθ is dense and 1 ||πθ (f )πθ (g) − πθ (f g + 2πθ{f, g})|| −→ 0. θ→0 θ In particular the last of the above mentioned facts leads to the interpretation of T2θ as a deformation of the group C*-algebra C ∗ (π1 (Z2 )) associated to the family of group 2-cocycles ωθ . We will use the above observation to introduce the following “minimal” requirements for a noncommutative deformation R of a Riemannian surface: • R is a unital (hopefully simple) C*-algebra; • both R and C( ) are fibers of a continuous family of C*-algebras over a path connected space; ∗ ( , σ) for some • R is (at least) KK-equivalent to a twisted group C*-algebra Cred group 2-cocycle σ.