FIXED-POINTS OF THE MAPPING CLASS GROUP IN THE SU(N) MODULI SPACES

Let Sigma be a compact oriented surface with or without boundary compo nents. In this note we prove that if chi(Sigma) < 0 then there exist i nfinitely many integers n such that there is a point in the moduli spa ce of irreducible flat SU(n) connections on Sigma which is fixed by an y orientation preserving diffeomorphism of Sigma, Secondly we prove th at for each orientation preserving diffeomorphism f of Sigma and each n greater than or equal to 2 there is some m such that f has a fixed p oint in the moduli space of irreducible flat SU(n(m)) connections on S igma. Thirdly we prove that for all n greater than or equal to 2 there exists an integer m such that the m'th power of any diffeomorphism fi xes a certain point in the moduli space of irreducible flat SU(n) conn ections on Sigma.