A uniqueness of periodic maps on surfaces

TL;DR

This paper extends Kulkarni's uniqueness result for periodic maps on surfaces of high genus, identifies new bounds for order-based uniqueness, and offers a concise proof of Wiman's theorem on maximal order.

Contribution

It generalizes the uniqueness phenomenon to higher genus surfaces with larger orders and provides a new proof of Wiman's classical theorem.

Findings

Uniqueness of periodic maps for genus > 30 with order > 8g/3

Existence of non-conjugate maps of order 8g/3 for arbitrarily large genus

Short proof of Wiman's theorem on maximal order 4g+2

Abstract

Kulkarni showed that, if g is greater than 3, a periodic map on an oriented surface S_g of genus g with order more than or equal to 4g is uniquely determined by its order, up to conjugation and power. In this paper, we show that, if g is greater than 30, the same phenomenon happens for periodic maps on the surfaces with orders more than 8g/3 and, for any integer N, there is g > N such that there are periodic maps of S_g of order 8g/3 which are not conjugate up to power each other. Moreover, as a byproduct of our argument, we provide a short proof of Wiman's classical theorem: the maximal order of periodic maps of S_g is 4g+2.