Non-commutative Two Dimensional Modular Symbol

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We define iteration over a two dimensional manifold as analog of iteration over a path defined by Chen. We give several applications. Some of them include constructions of non-abelian modular symbol for SL(3,) and for SL₂/ₖ, where K is a real quadratic field. We construct the latter as generating series of iterated integrals over a two-dimensional manifold of Hilbert modular forms. We give a motivic interpretation of this non-abelian modular symbol. Other applications give motivic interpretations of certain iterated completed Dedekind zeta functions.