Contractive representations of odometer semigroup

Given a natural number n ≥ 1, the odometer semigroup Oₙ, also known as the adding machine or the Baumslag-Solitar monoid with two generators, is a well-known object of study in group theory. This is also a central object of study for various algebraic-analytic structures. This paper examines the odometer semigroup in relation to representations of bounded linear operators. We focus on noncommutative operators and prove that contractive representations of Oₙ always admit to nicer isometric representations of Oₙ on vector-valued Fock spaces. We give a complete description of representations of Oₙ on the Fock space and relate it to the dilation theory and invariant subspaces of Fock representations of Oₙ. Along the way, we also classify Nica covariant representations of Oₙ.