Groups and Dynamics Seminar

Despite their elementary nature, one-relator groups entail a wide variety of behavior and so form a central testing ground of group theory. We focus on two orbit equivalence invariants of these groups, their cost and their first l2-Betti number $\beta_1^2(G)$. Gaboriau proved a general inequality of the form $cost(G)\ge \beta_1^2(G)+1$ for every countable infinite group G and asked whether equality always holds. In this talk, I will discuss Gaboriau's question for one-relator groups, crucially taking advantage of the fact that torsion-free one-relator groups are locally indicable. This is joint work with Antoine Poulin.