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Simon Machado: Discrete approximate subgroups of linear algebraic groups

Time: Wed 2021-05-12 13.15

Location: Zoom, meeting ID: 685 0671 8075

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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