Some applications of Diophantine Approximation to Group theory

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.

algebraic geometrynumber theoryrepresentation theory

Audience: researchers in the topic

Comments: Zoom ID: 854 7383 4027

Password: 562471

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POINTS - Peking Online International Number Theory Seminar

Series comments: Description: Seminar on number theory and related topics

This seminar series is sponsored by the Beijing International Center of Mathematical Research (BICMR) and the School of Mathematical Sciences of Peking University.

The conference number and password are available on the external website. See also the announcements on bicmr.pku.edu.cn

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