In fairly recent joint work with Corvaja, Rapinchuk, Ren, we applied results from Diophantine S-unit theory to problems of ‘bounded generation’ in linear groups: this property is a strong form of finite generation and is useful for several issues in the setting. Focusing on ‘anisotropic groups’ (i.e. containing only semi-simple elements), we could give a simple essentially complete description of those with the property. More recently, in further joint work also with Demeio, we proved the natural expectation that sets boundedly generated by semi-simple elements (in linear groups over number fields) are ‘sparse’. Actually, this holds for all sets obtained by exponential parametrizations. As a special consequence, this gives back the previous results with a different approach and additional precision and generality.
This video is part of the Number Theory Web Seminar series.
