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 at least 5/8, must be abelian. For infinite groups, one needs to work a bit harder to define the probability that a given law holds. One natural way is by sampling a random element uniformly from the r-ball in the Cayley graph and taking r to infinity; another way is by sampling elements using random walks. It was asked by Amir, Blachar, Gerasimova, and Kozma whether a law that holds with probability 1, must actually hold globally, for all elements. In a recent joint work with Be’eri Greenfeld, we give a negative answer to their question.

In this talk I will give an introduction to probabilistic group laws and present a finitely generated group that satisfies the law x^p=1 with probability 1, but yet admits no group law that holds for all elements.