On a Family of Representations of Residually Finite Groups

For a residually finite group G , its normal subgroups G ⊃ G 1 ⊃ G 2 ⋯ with ∩ n ∈ ℕ G n = { e } and for a growth function γ we construct a unitary representation π γ of G . For the minimal growth, π γ is weakly equivalent to the regular representation, and for the maximal growth it is weakly equivalent to the direct sum of the quasiregular representations on the quotients G / G n . In the case of intermediate growth we show two examples of different behaviour of π γ .