Can group laws be learned using random walks?

In various cases, a law that holds in a group with high probability must actually hold for all elements. For instance, a finite group in which the commutator law [x,y]=1 holds with probability larger than 5/8 must be abelian. In the talk I’ll discuss a probabilistic approach to laws on infinite groups, using random walks, and present results, joint with Be’eri Greenfeld, answering a few questions of Amir, Blachar, Gerasimova, and Kozma.