8 problems

Let $n$ be fixed and let $p$ be large. Consider Cayley graphs of $SL_2(\mathbb Z/p^n\mathbb Z)$ generated by random generating sets, together with the analogous graphs for the proj…

Let $n=2m$ with $n\geq 6$, and consider the Schreier coset graph of $S_n$ on binary strings with $m$ zeros and $m$ ones. Let $d_n$ be the distance from the specified starting verte…

A primitive origami is a square-tiled surface whose covering of the unit square torus does not factor through a nontrivial isogeny. The associated $\operatorname{SL}(2,\mathbb{Z})$…

Let $q$ be a prime number, let $S$ be a random subset of $\textit{GL}_k(\mathbb{Z}_q)$, and let $G_k$ be the Schreier graph of $S \subset \textit{GL}_k(\mathbb{F}_q)$ acting on…

Decidability equivalence. The following are equivalent:

Let $S_N$ act naturally on the set of $s$-tuples of distinct elements of $[N]=\{1,\ldots,N\}$, and choose independently and uniformly random permutations…

Let $B$ be the upper triangular Borel subgroup of $SL_2(\mathbb Z/p^2\mathbb Z)$, and consider the projective Schreier graph formed from the action of a generating set on the coset…

Bellaterra expander conjecture. The Bellaterra graphs $\{B_i\}_{i=1}^\infty$ are a family of two-sided expanders. The authors' computations suggest an eigenvalue gap of roughly…