Tobias Hartnick: Recent developments in infinite approximate group theory
The notion of an approximate group was coined by Terry Tao
in 2008. However, examples of approximate groups have appeared in
several different areas of mathematics over the last 50 years, in
particular in additive combinatorics, mathematical quasicrystals,
bounded cohomology and analytic group theory.
While a lot is known about the structure of finite approximate groups,
the structure of infinite approximate groups is still a mystery.
Recently, techniques from geometric group theory have been used to
obtain some insights into infinite approximate group theory, leading
to the notion of an approximate lattice.
We will discuss examples of approximate lattices and some of their key
structural features. In particular, we will explain the classification
of approximate lattices in abelian groups (due to current Abel prize
laureate Yves Meyer). We will then focus on some connections to
geometric group theory, diffraction theory and Kazhdan's Property (T).
Based on joint work with Michael Björklund.
