Random walk questions for linear quantum groups

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We study the discrete quantum groups Γ whose group algebra has an inner faithful representation of type π:C*(Γ)→ Mₖ(C). Such a representation can be thought of as coming from an embedding Γ⊂ Uₖ. Our main result, concerning a certain class of examples of such quantum groups, is an asymptotic convergence theorem for the random walk on Γ. The proof uses various algebraic and probabilistic techniques.