New Dimensional Estimates for Subvarieties of Linear Algebraic Groups

For every connected, almost simple linear algebraic group \(G\le \textrm{GL}_{n}\) $G\le {\text{GL}}_n$ over a large enough field K, every subvariety \(V\subseteq G\) $V\subseteq G$ , and every finite generating set \(A\subseteq G(K)\) $A\subseteq G(K)$ , we prove a general dimensional bound, that is, a bound of the form \( |A\cap V(\overline{K})|\le C_{1}|A^{C_{2}}|^{\frac{\dim (V)}{\dim (G)}} \) $$|A\cap V(\overline{K})|\le C_1{|A^{C_2}|}^{\frac{dim(V)}{dim(G)}}$$ with \(C_{1}\) $C_1$ , \(C_{2}\) $C_2$ depending only on n, \(\textrm{deg}(V)\) $\text{deg}(V)$ . The dependence of \(C_1\) $C_1$ on n (or rather on \(\dim (V)\) $dim(V)$ ) is doubly exponential, whereas \(C_2\) $C_2$ (which is independent of \(\textrm{deg}(V)\) $\text{deg}(V)$ ) depends simply exponentially on n. Bounds of this form have proved useful in the study of growth in linear algebraic groups since 2005 (Helfgott) and, before then, in the study of subgroup structure (Larsen–Pink: A a subgroup). In bounds for general V and G available before our work, the dependence of \(C_1\) $C_1$ and \(C_2\) $C_2$ on n was of exponential-tower type. We draw immediate consequences regarding diameter bounds for untwisted classical groups \(G(\mathbb {F}_{q})\) $G({\mathbb{F}}_q)$ . (In a separate paper, we derive stronger diameter bounds from stronger dimensional bounds we prove for specific families of varieties V.)