3 problems

Let $p$ and $q$ be distinct odd primes with $q>2\sqrt p$ and Legendre symbol $\left(\frac pq\right)=1$, let $G_q=\operatorname{PSL}_2(\mathbb{F}_q)$, and let $S_{p,q}\subset G_q$ b…

Coboundary expansion conjecture for random balanced Cayley complexes. For any fixed $k\geq 1$ there exist constants $C(k)<\infty$ and $\epsilon(k)>0$ such that for any group $G$, t…

Let $\mathcal{B}$ be the Bruhat–Tits building associated with $\mathrm{PGL}_{d}(F)$, where $F$ is a local field and $d\geq3$. For a finite quotient $X$ of $\mathcal{B}$, let…