2 problems

Let $\Gamma$ be a countable group acting measurably, essentially freely, and measure-preservingly on a probability space. The cost of such an action is the associated measurable-gr…

Let $\Gamma$ be a finitely generated group, and let $(\Gamma_i)$ and $(\Delta_i)$ be normal chains in $\Gamma$ with trivial intersection. For a chain $(\Gamma_i)$, define the rank…