C* exponential length of commutators unitaries in AH-algebras

For each unital C*-algebra A, we denote cel(CU). A/ D sup{cel(u) : u epsilon CU(A)}, where cel(u) is the exponential length of u and CU(A) is the closure of the commutator subgroup of U-0(A). In this paper, we prove that cel(CU) (A) >= 2 pi provided that A is an AH-algebra with slow dimension growth whose real rank is not zero. On the other hand, we prove that cel(CU)(A) <= 2 pi when A is an AH-algebra with ideal property and of no dimension growth (if we further assume that A is not of real rank zero, we have cel(CU)(A) = 2 pi).