# Simon Machado: Discrete approximate subgroups of linear algebraic groups

Time: Wed 2021-05-12 13.15

Location:

Lecturer: Simon Machado (University of Cambridge)

### Abstract

A symmetric subset $$A$$ of a group $$G$$ is called an approximate subgroup if there is a finite subset $$F$$ of $$G$$ such that the set of products of two elements $$A\cdot A$$ is contained in $$F \cdot A$$. Approximate subgroups appear naturally in various situations. For instance, finite approximate subgroups were studied as they are related to spectral gap considerations. Another interesting fact comes from Meyer’s work: he proved that discrete co-compact approximate subgroups of Euclidean spaces – the so-called mathematical quasi-crystals – are closely linked to Pisot numbers.

I will discuss two new results about discrete and infinite approximate subgroups of linear algebraic groups. First of all, in real algebraic soluble groups the coarse structure is quite simple : approximate subgroups are coarsely equivalent to Zariski-closed normal subgroups. This leads to a first extension of Meyer’s results to all soluble Lie groups.

I will then consider other potential extensions. For such purposes Björklund and Hartnick defined strong approximate lattices - a certain non-commutative generalisation of mathematical quasi-crystals. I will explain how ergodic theoretic considerations show that any strong approximate lattice in $$SL_n(R)$$ with $$n \geq 3$$ is obtained as the set of matrices with coefficients belonging to the set of Pisot numbers of some number field $$K$$. This is an extension of both Meyer’s theorem and Margulis’ arithmeticity theorem.

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