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In this work we study state-closed representation of metabelian groups of wreath type, with emphasis on the Lamplighter groups of the type Gp,d = Cp o C. This study was motivated by a particular state-closed representation of degree 2 of the Lamplighter group C2 o C, which was used to determine the spectrum of C2 o C as a group of linear operators and thus give a counterexample to a conjecture of Atiyah. In the case d = 1, we characterize the state-closed representations of degree p of the groupGp,1. For the case d > 1, we show the groupGp,d has a stateclosed representation of degree p, but does not have a state-closed representation of degree q, where q is prime number. Furthermore, we prove the representation obtained for G2,2 is finite-state.

Representações fechadas por estado de grupos metabelianos tipo entrelaçado