Generating the liftable mapping class groups of regular cyclic covers

Let Mod(Sg) be the mapping class group of the closed orientable surface of genus g ≥ 1 , and let LModₚ(X) be the liftable mapping class group associated with a finite-sheeted branched cover p:S → X , where X is a hyperbolic surface. For k ≥ 2 , let pₖ: Sk(g-1)+1 → Sg be the standard k -sheeted regular cyclic cover. In this paper, we show that ₖ(Sg) ≥ 2 forms an infinite family of self-normalising subgroups in Mod(Sg) , which are also maximal when k is prime. Furthermore, we derive explicit finite generating sets for LModpₖ(Sg) for g ≥ 3 and k ≥ 2 , and LModp₂(S₂) . For g ≥ 2 , as an application of our main result, we also derive a generating set for LModp₂(Sg) ∩ CMod(Sg)(ι) , where CMod(Sg)(ι) is the centraliser of the hyperelliptic involution ι ∈ Mod(Sg) . Let L be the infinite ladder surface, and let qg : L → Sg be the standard infinite-sheeted cover induced by hᵍ⁻¹ where h is the standard handle shift on L . As a final application, we derive a finite generating set for LModqg(Sg) for g ≥ 3 .