4 problems

Conjecture on the non-prime density. The logarithmic density satisfies

Let $\Gamma$ be a Fuchsian group, and let $G$ be a finite simple classical group. A randomly chosen homomorphism in ${\rm Hom}(\Gamma,G)$ is an epimorphism with a probability that…

Let $(G_n)$ be any sequence of finite simple groups such that $|G_n|$ tends to infinity with $n$. Let $\mathbb{P}_2(G)$ denote the probability that two randomly chosen elements gen…

Let $G$ be a group, let $\{\mu_n\}_{n\in \mathbb{N}}$ be a sequence of “reasonable” measures on $G$, and let $\mathcal{E}\subseteq F_k$ be a set of equations in $k$ variables. The…