Lawrence and Sawin have shown that up to translation, any abelian variety over a number field contains only finitely many smooth ample hypersurfaces with given fundamental class and good reduction outside a given finite set of primes. A key ingredient in their proof is that certain Tannaka groups attached to smooth hypersurfaces are big. In the talk I will give a general introduction to Tannaka groups of perverse sheaves on abelian varieties and explain how to determine them for subvarieties of higher codimension (this is work in progress with Ariyan Javanpeykar, Christian Lehn and Marco Maculan).