We introduce block maps for subfactors and study their dynamic systems. We prove that the limit points of the dynamic system are positive multiples of biprojections and zero. For the \mathbb{Z}_2 case, the asymptotic phenomenon of the block map coincides with that of the 2D Ising model.The study of block maps requires a further development of our recent work on the Fourier analysis of subfactors.We generalize the notion of sum set estimates in additive combinatorics for subfactors and prove the exact inverse sum set theorem. Using this new method, we characterize the extremal pairs of Young's inequality for subfactors, as well as the extremal operators of the Hausdorff-Young inequality.

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The diameter of a disc filling a loop in the universal covering of a Riemannian manifold M may be measured extrinsically using the distance function on the ambient space or intrinsically using the induced length metric on the disc. Correspondingly, the diameter of a van Kampen diagram
filling a word that represents the identity in a finitely presented group 􀀀 can either be measured intrinsically in the 1–skeleton of
or extrinsically in the Cayley graph of 􀀀. We construct the first examples of closed manifolds M and finitely presented groups 􀀀 = π1M for which this choice — intrinsic versus extrinsic — gives rise to qualitatively different min–diameter filling functions.

We prove that the higher harmonic signature of an even dimensional oriented Riemannian foliation F of a compact Riemannian manifold M with coefficients in a leafwise U(p, q)-flat complex bundle is a leafwise homotopy invariant. We also prove the leafwise homotopy invariance of the twisted higher Betti classes. Consequences for the Novikov conjecture for foliations and for groups are investigated.

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