6 problems

Let $\Gamma$ be an infinite group with Kazhdan property (T), and let $(\Gamma_n)$ be a normal chain in $\Gamma$ with trivial intersection. Write $\mathrm{RG}(\Gamma,(\Gamma_n))$ fo…

Let $\Gamma$ be a lattice in a higher-rank Lie group, and let its absolute rank gradient be the infimum of $(d(H)-1)/[\Gamma:H]$ over finite-index subgroups $H\leq\Gamma$, where…

Let $\Gamma$ be a finitely generated residually finite group satisfying a non-trivial group law. The rank gradient $RG(\Gamma)$ measures the asymptotic normalized growth of the min…

Lackenby's conjecture. For every real number $\alpha>0$ there exists a finitely generated group $\Gamma$ such that

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…

Let $M$ be a finite-volume hyperbolic $3$-manifold and let $\{M_i\longrightarrow M\}$ be a family of finite-sheeted covers. Write $\operatorname{rgr}(M,\{M_i\})$ for the rank gradi…