Abstract

An approximate group is a subset of a group that is “almost closed” under multiplication. Finite approximate subgroups play a major role in additive combinatorics. In 2012, Breuillard, Green and Tao established a structure theorem concerning finite approximate subgroups and used this theory to reprove Gromov’s polynomial growth theorem. Infinite approximate groups were studied implicitly long before the formal definition. Meyer, in 1972, developed a theory of mathematical quasi-crystals which can be seen as approximate subgroups of R^n that are Delone. Recently, Björklund and Hartnick have begun a program investigating infinite approximate lattices in locally compact second countable groups using geometric and measurable structures. In the talk I will introduce infinite approximate groups and their geometric aspects. This is joint work with Hartnick and Tonić.