Homology Covers of Riemann Surfaces and Automorphisms: Examples

If either k ≥ 2 is an integer or k=∞, then associated to a Riemann surface S is its k-homoly cover Sₖ. The surface Sₖ has a group of conformal automorphisms Mₖ H₁(S; Zₖ), where Z∞= Z, such that S=Sₖ/Mₖ. Moreover, for every group L ≤ Aut(S), there is a short exact sequence 1 → Mₖ → Lₖ → L → 1, where Lₖ is a subgroup of conformal automorphisms of Sₖ. If k ≠ ∞ and S is analytically finite and non-exceptional, then Mₖ, Lₖ and L are finite groups. In this case, if k and $|L|$, the order of L, are relatively prime integers, then the Schur-Zassenhaus theorem asserts that Lₖ Mₖ L. In this paper, we investigate situations when k and $|L|$ are not necesarily relatively primes.