Epimorphic subgroups of simple algebraic groups

TL;DR

The paper addresses the problem of identifying small, proper epimorphic subgroups within simple algebraic groups over algebraically closed fields of positive characteristic. It develops a general BB-method framework using rank-one subgroups $J\cong A_1$ and 1-dimensional unipotent subgroups $Y$ normalized by Borel subgroups of $J$, enabling the construction of epimorphic subgroups $H=B_JY$ with controlled dimensions. Through a detailed case analysis across classical and exceptional types, the authors produce explicit epimorphic subgroups of dimensions $3$, $4$, or at most $5$, with a bound of $\dim H\le 5$ guaranteed; in many cases the dimension can be reduced to $3$ or $4$, and in characteristic $p\ge h(G)$ one can realize $\dim H=3$. The results have geometric significance for constructing generically rationally connected homogeneous spaces and illustrate how epimorphic subgroups can be realized uniformly across types, characteristics, and Coxeter number bounds. The work thus provides concrete, small-dimension epimorphic witnesses that complement existing structure theorems and extend the toolkit for applications in algebraic geometry and representation theory.

Abstract

A morphism of linear algebraic groups $φ:K\rightarrow G$ is called an epimorphism if it admits right cancellation. A subgroup $H\leq G$ is epimorphic if the inclusion map is an epimorphism. For $G$ a simple algebraic group over an algebraically closed field of arbitrary characteristic we construct epimorphic subgroups of bounded dimension (at most five).

Theorems & Definitions (13)

• Theorem 1
• Theorem 2
• Lemma 2.1
• Proposition 2.2
• proof
• proof : Proof of Theorem \ref{['mainthm2']}
• Lemma 3.1
• proof
• Proposition 3.2
• proof