2 problems

Let $M$ be a closed hyperbolic $3$-manifold. A subgroup of $\pi_1 M$ is quasi-Fuchsian if it is the fundamental group of a quasi-Fuchsian immersed surface in $M$, and a subgroup is…

Let $F$ be a finitely generated free group. For $g_1,g_2,\ldots,g_k\in F$, write $[g_i]^F$ for the conjugacy class of $g_i$ in $F$, let $N(g_1,g_2,\ldots,g_k)$ be the normal subgro…