3 problems

Let $G$ be a finitely generated perfect group, meaning $G=[G,G]$, the commutator subgroup of $G$. Wiegold's normal generation conjecture. $G$ can be normally generated by a single…

Let $G$ be a torsionfree discrete group, and let $g_1,\dots,g_k$ be elements whose normal closure is $G$. The first $\ell^2$-Betti number of $G$ is denoted by $\beta_1^{(2)}(G)$. N…

Let $G$ be a torsion free discrete group, or a $p$-group for a prime $p$, and let $g_1,\dots,g_n$ be arbitrary elements of $G$. Denote by…