On a family of representations of the Higman–Thompson groups

Abstract

Abstract We obtain an uncountable family of inequivalent and irreducible representations of the Higman–Thompson groups F n ⊂ T n ⊂ V n F_{n}\subset T_{n}\subset V_{n} . This is accomplished by considering a family of representations of the Higman–Thompson groups V n V_{n} that arise from representations of Cuntz algebras, each one acting on a Hilbert space built upon the orbit of a point x ∈ [ 0 , 1 ) x\in[0,1) under the dynamical system Φ ⁢ ( x ) = n ⁢ x ( mod 1 ) \Phi(x)=nx\pmod{1} . Every such representation is retrieved through the action of V n V_{n} on orb ⁡ ( x ) \operatorname{orb}(x) , and their restrictions to the subgroups F n F_{n} and T n T_{n} of V n V_{n} are studied using properties of the groups.