Characterizing a vertex-transitive graph by a large ball

The subject of the talk will be vertex-transitive infinite connected graphs with bounded degree and with a property of large scale simple connectedness. The most classical examples of such objects are Cayley graphs of finitely presented groups, but I will explain that there are (uncountably) many other, and I will study some topological questions on the space of all such graphs. In particular I will give some answers to a question of Benjamini and Georgakopoulos asking which Cayley graphs are isolated. Based on joint works with Romain Tessera.