On the growth of Fourier multipliers

We define a sequence of functions, namely, tame cuts, in the Fourier algebra A(G) of a locally compact group G, which satisfies certain convergence and growth conditions. This new consideration allows us to give a group admitting a Fourier multiplier that is not completely bounded. Furthermore, we show that the induction map MA(F) ! MA(G) is not always continuous. We also show how Liao's property (TSchur, G, K) opposes tame cuts. Some examples are provided.