Maximum orders of periodic maps on closed surfaces

AbstractIn this paper, we determine the maximum orders of periodic maps on closed surfaces. For the orientable closed surface of genus g, g⩾2, this maximum order is 4g + 2 (realized by an orientation preserving map) if g is odd and is 4g + 4 (realized by an orientation reversing map) otherwise. For the nonorientable closed surface of genus q, q⩾3, this maximum order is 2q if q is odd and is 2q − 2 otherwise. Furthermore, the maximum order of orientation reversing periodic maps on Fg is 4g − 4 if g is odd