We show that if A is a simple Villadsen algebra of either the first type with seed space a finite dimensional CW complex, or of the second type, then A absorbs the Jiang-Su algebra tensorially if and only if the central sequence algebra of A does not admit characters. Additionally, in a joint appendix with Joan Bosa (see the following paper), we show that the Villadsen algebra of the second type with infinite stable rank fails the Corona Factorization Property, thus providing the first example of a unital, simple, separable and nuclear C^*-algebra with a unique tracial state which fails to have this property.