Applications of Diophantine Approximation in Group Theory
Dr. Jinbo Ren
2022-06-14 10:30-11:30
Online
Speaker: Dr. Jinbo Ren (IAS)Time: 10:30-11:30 June 14, 2022 (Tuesday)
Place: Online (Zoom ID: 4663562952 Password: mcm1234)
Title: Applications of Diophantine Approximation in Group Theory
Abstract: An abstract group is said to have the {\it bounded generation} property (BG) if it can be written as a product of finitely many cyclic subgroups. Being a purely combinatorial notion, bounded generation has close relation with many group theoretical problems including semi-simple rigidity, Kazhdan's property (T) and Serre's congruence subgroup problem.
In this talk, I will explain how to use the Schlickewei-Schmidt subspace theorem in Diophantine approximation to prove that the distribution of the points of a set of matrices over a number field $F$ with (BG) by certain fixed semi-simple (diagonalizable) elements is of at most logarithmic size when ordered by height. Moreover, one obtains that a linear group $\Gamma \subset \mathrm{GL}_n(K)$ over a field $K$ of characteristic zero admits a {\it purely exponential parametrization} if and only if it is finitely generated and the connected component of its Zariski closure is a torus.
This is joint work with Corvaja, Demeio, Rapinchuk and Zannier.
