Local properties of groups
8th December 2021, 3:00 pm – 4:00 pm
Physics, Powell Lecture Theatre
Is commutativity a ‘global’ or a ‘local’ property of a finitely generated group G? On the one hand, the classification of finitely generated abelian groups imposes rather stringent conditions on the global structure of G, suggesting we should think of commutativity as a ‘global’ property. On the other hand, if we endow G with the word metric corresponding to a given symmetric generating set S (so the distance from an element g to the identity e is the minimum number of elements of S needed to express g), then one can tell whether G is commutative by examining only the ball of radius 2 centred at e. In this sense, one could reasonably describe commutativity as a ‘local’ property.
In this talk I will not attempt to give a precise definition of what it means for a group property to be ‘local’ or ‘global’. What I will do is present a number of examples where taking what might be called a ‘local’ approach to ‘global’ questions and structures in group theory can yield a sometimes surprising amount of additional understanding. In particular, I will discuss a certain ‘local’ version of the notion of a quotient group, as recently developed in joint work with Tom Hutchcroft, and describe an application to percolation on finite transitive graphs. I will also discuss a ‘local’ version of polynomial growth appearing in celebrated work of Breuillard, Green and Tao on approximate groups, and talk about some applications to random walks.