Applications of the Subspace Theorem in Group Theory

Jinbo Ren, IAS
Fine Hall 214

In-Person and Online Talk 

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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 semisimple 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.