On the topological type of anticonformal square roots of automorphisms of even order of Riemann surfaces (Q6632410)

Two automorphism groups \(G_1\) and \(G_2\) of a compact Riemann surface \(S\) of genus \(\sigma\) are said to be topologically equivalent if there is a homeomorphism \(h \colon S\rightarrow S\) such that \(G_2=hG_1h^{-1}\). The topological types of automorphism groups of surfaces of genus \(\sigma\) provide a stratification of the moduli space of compact Riemann surfaces of genus \(\sigma\) called the equisymmetric stratification. A first step in describing this stratification is the classification of topological types of cyclic groups of automorphisms. In the case when an automorphism \(f\) is conformal, a result of \textit{J. Nielsen} [Danske Vidensk. Selsk. Math.-fys. Medd. 15, Nr. 1, 77 s (1937; JFM 63.0553.03)] states that the topological type is completely determined by its order and the local behavior, specifically, the rotational angles at the fixed points of \(f\). Topological classification of anticonformal automorphisms typically depends on more invariants and so is more difficult. In the paper under review, the author develops conditions for when certain anticonformal automorphisms are topologically equivalent.\N\NSuppose that \(f\) is a conformal automorphism of a compact Riemann surface \(S\) of even order \(m\), let \(g_1\) and \(g_2\) be anticonformal square roots of \(f\) (meaning \(g_1^2=g_2^2=f\)) and suppose that the order of \(\langle g_1,g_2\rangle\) is \(2nm\). The main result of the paper are sufficient conditions for when \(\langle g_1 \rangle\) and \(\langle g_2 \rangle\) are topologically equivalent. Specifically, when \(n\) is odd, \(\langle g_1 \rangle\) and \(\langle g_2 \rangle\) are always topologically equivalent. The case when \(n\) is even is a little more complicated and depends on the genus of \(S/\langle g_1,g_2\rangle\). Specifically, when the genus of \(S/\langle g_1,g_2\rangle\) is even and the genus of neither \(S/\langle g_1\rangle\) or \(S/\langle g_2\rangle\) is \(2\), or when the genus of \(S/\langle g_1,g_2\rangle\) is odd and \(\langle g_1,g_2\rangle\) is Abelian, then \(\langle g_1 \rangle\) and \(\langle g_2 \rangle\) are topologically equivalent. Moreover, the author provides explicit examples to show that when these conditions are relaxed, there are examples of anticonformal square roots of \(f\) that are not topologically equivalent.\N\NThe results are developed using uniformization of Riemann surfaces and the correspondence it provides between automorphism groups and Fuchsian and NEC groups.