加入Zoom 会议
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Abstract: Transcendental Number Theory tells us an essential difference between transcendental numbers and algebraic numbers is that the former can be approximated by rational numbers ``very well’’ but not the latter. More specifically, one has the following Fields Medal work by Roth. Given a real algebraic number $a$ of degree $\geq 3$ and any $\delta>0$, there is a constant $c=c(a,\delta)>0$ such that for any rational number $\eta$, we have $|\eta-a|>c H(\eta)^{-\delta}$, where $H(\eta)$ is the height of $\eta$. Moreover, we have Schmidt’s Subspace theorem, a non-trivial generalization of Roth’s theorem.
On the other hand, we have the notion of Bounded Generation in Group Theory. An abstract group $\Gamma$ is called Boundedly Generated if there exist $\g_1,g_2,\cdots, g_r\in \Gamma$ such that $\Gamma=\langle g_1\rangle \cdots \langle g_r\rangle$ where $\langle g\rangle$ is the cyclic group generated by $g$. While being a purely combinatorial property of groups, bounded generation has a number of interesting consequences and applications in different areas. For example, bounded generation has close relation with Serre’s Congruence Subgroup Problem and Margulis-Zimmer conjecture.
In my recent joint work with Corvaja, Rapinchuk and Zannier, we applied an ``algebraic geometric’’ version of Roth and Schmidt’s theorems, i.e. Laurent’s theorem, to prove a series of results about when a group is boundedly generated. In particular, we have shown that a finitely generated anisotropic linear group over a field of characteristic zero has bounded generation if and only if it is virtually abelian, i.e. contains an abelian subgroup of finite index.
In my talk, I will explain the idea of this proof and give certain open questions.
摘要：超越数论中一个基本结论是说，超越数和代数数的一个关键区别在于前者可以被有理数“很好地”逼近，而后者不能。特别地，作为Roth的菲尔兹奖工作，他证明了如下结论。给定一个次数至少为$3$的实代数数 $a$，则对任意$\delta>0$，存在常数$c=c(a,\delta)>0$ 使得对任意有理数$\eta$，我们总有$|\eta-a|>c H(\eta)^{-\delta}$。这里$H(\eta)$是有理数$\eta$的高度。进一步地，作为Roth定理的非平凡推广，我们有Schmidt的子空间定理。
另一方面，在群论中，一个抽象群$\Gamma$被称为是有界生成的，如果存在$g_1,g_2,\cdots, g_r\in \Gamma$ 使得$\Gamma=\langle g_1 \rangle \cdots \langle g_r\rangle$，其中$\langle g\rangle$是由$g$生成的循环群。有界生成这个概念和Serre的同余子群问题以及Margulis-Zimmer猜想等主题密切相关。
在Corvaja, Rapinchuk, Zannier和我的一项合作中，我们使用Roth和Schmidt的工作的一个更广泛的代数几何版本，即Laurent定理，来给出一个抽象群是否为有界生成的若干刻画。特别地，我们证明了一个定义在特征为$0$的域上的各向异性的线性群是有界生成的当且仅当它是有限生成的且“几乎”是阿贝尔群，即包含一个指数有限的阿贝尔子群。
在报告中，我会向大家介绍证明的大致思路并给出若干开放问题。
