8 problems

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 $G=\mathbb{Z}_n$ with $n$ prime, and fix the number of random generators $k$. Let $\operatorname{diam}(G_k)$ denote the diameter of the random Cayley graph. Diameter limit conj…

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

Let $w$ be a group law, and consider the probability that independent random walks satisfy $w$ on a finitely generated group. The preceding question asks whether the corresponding…

Let $G$ be an infinite group, and suppose that independent random-walk samples satisfy the law $x^2=1$ with positive probability. Leemann–de la Salle conjecture. If $G$ satisfies…

Let $G_i$ be a sequence of finite simple groups with $|G_i|\to\infty$, and suppose that the primes $r$ and $s$ both divide $|G_i|$. Let $P_{r,s}(G_i)$ denote the probability that a…

For primes $r,s$, let $P_{r,s}(G)$ be the probability that a uniformly random pair of elements of orders $r$ and $s$ generates the finite simple group $G$, and let $L_{r,s}$ be the…

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…