Low-genus primitive monodromy groups with a nonunique minimal normal subgroup

TL;DR

The paper addresses the classification of indecomposable degree-$n$ coverings $f:X\to \mathbb{P}^1_{\mathbb{C}}$ whose monodromy group $G\le S_n$ is a primitive group of type-B (having two nonabelian minimal normal subgroups). It combines the Aschbacher--O'Nan--Scott framework, fixed-point-ratio bounds, and Riemann–Hurwitz constraints to prune the landscape of genus-$g$ systems, culminating in a unique genus-$1$ example with $n=168$ and $G\cong {\rm PSL}_2(7)^2.C_2$ (ramified with type $(2,3,8)$, two variants). The results imply that for $g_X\le 1$, there is exactly one such covering, and for arbitrary $g$ there are no such coverings when $n$ is sufficiently large. This work completes the genus-$0$/genus-$1$ classification for type-B primitive monodromy groups, extending prior observations and validating the broader genus-$g$ program via a combination of theoretical bounds and computational verification (GAP/MAGMA).

Abstract

Let $X$ be a Riemann surface, and let $f:X\to\mathbb{P}^1_\mathbb{C}$ be an indecomposable (branched) covering of genus $g$ and degree $n$ whose monodromy group has more than one minimal normal subgroup. Closing a gap in the literature, we show that there is only one such covering when $g\leq 1$. Moreover, for arbitrary $g$, there are no such coverings with $n\gg_g 0$ sufficiently large.

Theorems & Definitions (21)

• Theorem 1
• Lemma 2.1: Aschbacher_1990
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• Proposition 3.1
• proof
• Proposition 3.2
• proof