In this talk I will present a useful construction introduced by Kassabov and Pak, called diagonal products. They arise naturally in the study of the space of marked groups, that is, the normal subgroups of a free group with the Chabauty topology. It turns out that diagonal products are extremely rich, and proved to be a useful tool for providing a full spectrum of various growth functions for groups. This includes answers to questions such as "How fast do Følner sets grow in an amenable group" and "How fast do residual chains grow in residually finite groups" etc. We will elaborate on a joint work with Arie Levit and Itamar Vigdorovich, where we give a general classification result for characters (á la Thoma) of many diagonal products. As a result, we deduce that there are uncountably many Hilbert–Schmidt stable groups, which are as unstable as one wants.
