WoG 2023 Talk 5.5: Ismael Morales - Virtual homological invariants and profinite rigidity

Speaker: Ismael Morales (U. Oxford)

Title: Virtual homological invariants and profinite rigidity
Abstract: The virtual second betti number of a free group (resp. surface group) is equal to zero (resp. one). We will discuss the conjecture that these elementary properties should, in fact, characterise such groups among finitely generated residually finite groups. We also provide some evidence for these conjectures, since they are true among limit groups and hyperbolic fundamental groups of graphs of free groups with cyclic edge subgroups. Finally, we relate these conjectures with several questions on profinite rigidity. This is based on joint work with Jonathan Fruchter.