New dimensional estimates for subvarieties of linear algebraic groups
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912055960797184 |
|---|---|
| author | Bajpai, Jitendra Dona, Daniele Helfgott, Harald Andrés |
| author_facet | Bajpai, Jitendra Dona, Daniele Helfgott, Harald Andrés |
| contents | For every connected, almost simple linear algebraic group $G\leq\mathrm{GL}_{n}$ over a large enough field $K$, every subvariety $V\subseteq G$, and every finite generating set $A\subseteq G(K)$, we prove a general dimensional bound, that is, a bound of the form \[|A\cap V(\overline{K})|\leq C_{1}|A^{C_{2}}|^{\frac{\dim(V)}{\dim(G)}}\] with $C_{1},C_{2}$ depending only on $n,\mathrm{deg}(V)$. The dependence of $C_1$ on $n$ (or rather on $\dim (V)$) is doubly exponential, whereas $C_2$ (which is independent of $\mathrm{deg}(V)$) 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 $C_1$ and $C_2$ on $n$ was of exponential-tower type. We draw immediate consequences regarding diameter bounds for untwisted classical groups $G(\mathbb{F}_{q})$. (In a separate paper, we derive stronger diameter bounds from stronger dimensional bounds we prove for specific families of varieties $V$.) |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_09197 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | New dimensional estimates for subvarieties of linear algebraic groups Bajpai, Jitendra Dona, Daniele Helfgott, Harald Andrés Group Theory Algebraic Geometry Combinatorics Primary: 20F69, 20G40, 05C25, Secondary: 14A10, 05C12 For every connected, almost simple linear algebraic group $G\leq\mathrm{GL}_{n}$ over a large enough field $K$, every subvariety $V\subseteq G$, and every finite generating set $A\subseteq G(K)$, we prove a general dimensional bound, that is, a bound of the form \[|A\cap V(\overline{K})|\leq C_{1}|A^{C_{2}}|^{\frac{\dim(V)}{\dim(G)}}\] with $C_{1},C_{2}$ depending only on $n,\mathrm{deg}(V)$. The dependence of $C_1$ on $n$ (or rather on $\dim (V)$) is doubly exponential, whereas $C_2$ (which is independent of $\mathrm{deg}(V)$) 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 $C_1$ and $C_2$ on $n$ was of exponential-tower type. We draw immediate consequences regarding diameter bounds for untwisted classical groups $G(\mathbb{F}_{q})$. (In a separate paper, we derive stronger diameter bounds from stronger dimensional bounds we prove for specific families of varieties $V$.) |
| title | New dimensional estimates for subvarieties of linear algebraic groups |
| topic | Group Theory Algebraic Geometry Combinatorics Primary: 20F69, 20G40, 05C25, Secondary: 14A10, 05C12 |
| url | https://arxiv.org/abs/2308.09197 |