Subspace stabilizers and maximal subgroups of exceptional groups of Lie type

In 1998, Liebeck and Seitz introduced a constant $t(G)$, dependent on the root system of a reductive algebraic group $G$ and proved that if $x$ is a semisimple element of order greater than $t(G)$ in $G$ then there exists an infinite subgroup of $G$ stabilizing the same subspaces of $L(G)$ as $x$. The values for $t(G)$ are $12$, $68$, $124$ and $388$ for $G=G_2,F_4,E_6,E_7$ respectively. In this paper we obtain a similar result for these groups and the minimal module $V_{\mathrm{min}}$, obtaining significantly smaller numbers, namely $4$, $18$, $27$ and $75$ respectively (with some small conditions on the element $x$ that are not important for applications). Note that both $t(G)$ and these new bounds are sharp. As a corollary we eliminate several potential maximal subgroups $\mathrm{PSL}_2(q_0)$ of these groups that seem difficult to eliminate through other means, along with other groups. This paper forms part of the author's programme to vastly reduce the number of putative maximal subgroups of exceptional groups of Lie type.