Minimally weakly almost periodic groups

A noncompact locally compact group G is minimally weakly almost periodic (w.a.p.) if each w.a.p. function on G can be written as g + h where g is almost periodic and h vanishes at infinity. If G is minimally w.a.p., then the coefficient functions of unitary representations which have no nonzero finite-dimensional subrepresentations vanish at infinity. We show that M(n), the motion group of the Euclidean n-space, and SL(2, R) are minimally w.a.p. We also show that a noncompact locally compact connected solvable group G is minimally w.a.p. if and only if G K(G) ≅ M(2) where K(G) is the largest compact normal sub-group of G.