Odd-Order Group Actions on Surfaces

Let *() denote the order of the largest odd-order group of automorphisms that a compact Riemann surface of genus can admit. In this paper we obtain sharp upper and lower bounds for *(), and exhibit infinite sequences of genera in which the upper bound is attained and infinite sequences in which it is not attained. We show that the lower bound is attained at least once, and that for infinitely many , it is the order of the largest odd-order cyclic group of automorphisms that a surface of genus can admit. We exhibit infinite sequences of genera which admit “large” odd-order actions by certain semi-direct products of cyclic groups