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### p-Adicizable Discrete Variants of Classical Lie Groups & Coset Spaces in TGD Framework

#### Abstract

*p*-Adization of quantum TGD is one of the long term projects of TGD. The notion of finite measurement resolution reducing to number theoretic existence in

*p*-adic sense is the fundamental notion.

*p*-Adic geometries replace discrete points of discretization with

*p*-adic analogs of monads of Leibniz making possible to construct differential calculus and formulate p-adic variants of field equations allowing to construct

*p*-adic cognitive representations for real space-time surfaces. This leads to a construction for the hierarchy of

*p*-adic variants of imbedding space inducing in turn the construction of

*p*-adic variants of space-time surfaces. Number theoretical existence reduces to conditions demanding that all ordinary (hyperbolic) phases assignable to (hyperbolic) angles are expressible in terms of roots of unity (roots of

*e*). The construction reduces to the construction of

*p*-adicizable discrete subgroups of classical Lie groups. The construction starts from

*SU*(2) and

*U*(1) and proceeds iteratively. Remarkably, the finite discrete

*p*-adicizable subgroups of

*SU*(2) correspond to those appearing in the hierarchy of inclusions of hyperfinite factors and include the groups assignable to Platonic solids. One can see them as cognitively especially simple finite

*p*-adicizable objects providing

*p*-adic approximation of sphere. The Platonic solids have analogs also for larger classical Lie groups.