7 problems

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…

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 $G$ be an infinite, residually finite, boundedly generated group. Its rank gradient is … where $H$ ranges over the finite-index subgroups of $G$. Abert–Jaikin-Zapirain–Nikolov'…

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

Let $K$ be a number field, let $p$ be a prime, and let $S$ be a set of places such that the Galois group $G_{K,p,S}$ is infinite and the hypotheses of the source's theorem on this…

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…