4 problems

Let $p$ be a prime, and let $f:\mathbb{N}\rightarrow\mathbb{N}$ be any function. A pro-$p$ group $\Gamma$ is finitely presented if it admits a finite presentation in the pro-$p$ ca…

Let $\Gamma$ be the fundamental group of a connected, orientable, irreducible, compact 3-manifold whose boundary is empty or a collection of tori. For finite-index subgroups…

Let $N$ be a prime 3-manifold, and let $\operatorname{vol}(N)$ denote the sum of the volumes of the hyperbolic pieces in its geometric decomposition. Write $\operatorname{Tor} A$ f…

Let $L$ be a collection of $m>1$ linked solid tori in the three-sphere, let $M=\mathbb{S}^3-L$, and let $G=H_1(M)\cong\mathbb{Z}^m$. For each finite-index subgroup $H\subset G$, le…