The rank-gradient chain-independence conjecture

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 gradient by

$$\mathrm{RG}(\Gamma,(\Gamma_i))=\lim_{i\rightarrow\infty}\frac{d(\Gamma_i)-1}{[\Gamma:\Gamma_i]},$$

where $d(\Gamma_i)$ is the minimal number of generators of $\Gamma_i$. Rank-gradient chain-independence conjecture.

$$\mathrm{RG}(\Gamma,(\Gamma_i))=\mathrm{RG}(\Gamma,(\Delta_i)).$$

The conjecture asks whether rank gradient depends only on the group rather than on the chosen normal chain. It is related to measurable group actions and cost, and the source describes it as an open problem.