16 problems

Cigler's conjecture. If a permutation-like matrix group contains a maximal cycle, then it is a permutation matrix group.

Kim–Koberda conjecture. For every $q\in\mathbb{Q}\cap(-4,4)$, the group $G_q$ is not freely generated by $A$ and $B_q$. When $|q|\geq 4$, a ping-pong argument shows that $G_q$ is d…

For $\lambda\in\mathbb{C}$, let … Call $\lambda$ free if $G_{\lambda}$ is a free group, and non-free otherwise. Kim–Koberda conjecture. Every $\lambda\in\mathbb{Q}\cap(-2,2)$ is no…

Bounded non-scalar length conjecture. For every $n$ there exists a number $f_n$ such that every spherical equation over $\operatorname{GL}(n,p)$ containing at least $f_n$ non-scala…

Polynomial-time triangular-matrix conjecture. The restriction to prime moduli is unnecessary, and

Constant-norm geodesic conjecture. Shifted exponential solutions of the form $Be^{tC}$ are the only geodesics with constant norm when viewed as curves in the vector space…

Nonexistence conjecture. If $n$ is odd, then $\mathsf{SL}(n)$ has no bounded geodesics.

Let $q=p^r$ be any prime power, and let ${\mathbb{F}}_q$ be the corresponding finite field. Consider the Gill–Helfgott result for subsets $A\subseteq GL_n({\mathbb{F}}_p)$ whose ge…

Let $n\ge 1$ be an integer and let $G=\operatorname{SL}_n(\mathbb{F}_p)$. Consider the solutions $(g,h)\in G\times G$ of … for a fixed $x\in G$. Hu–Li's conjecture. The solutions s…

For $N>0$, let $\|\cdot\|$ be the matrix norm and count quadruples of matrices $\g_1,\dots,\g_4\in\operatorname{SL}_{2}(\mathbb Z)$ with $\|\g_j\|<N$ satisfying an additive relatio…

Let $G$ be a group of $n \times n$ tropical matrices with real entries. Tropical matrix group conjecture. The group $G$ has a torsion-free abelian subgroup of index at most $n!$. T…

Let $Q$ be a division ring and let $S=\mathrm{M}_n(Q)$ be the monoid of $n\times n$ matrices over $Q$, with $n\geqslant 3$. Let $E=E(S)$, and consider the free idempotent generated…

Three-dimensional Fibonacci-operator conjecture. Every best approximation other than $\mathcal A_3$ belongs to the set

Let $B$ be the non-hyperbolic operator displayed in the source, and let $(1,0,0)$ be the initial integer vector. The best approximations are considered through the points…

Let $F_k$ denote the $k$-th Fibonacci number, and consider the MCRS-group containing the Fibonacci matrix. For a positive integer $N$, choose $k$ so that … The two eigenspaces in q…

Let $\mathcal A_N\in\Omega$ be the best approximation to an algebraic MCRS-group at a positive integer $N$, and let $v_1$ and $v_2$ be the primitive vectors defining $\mathcal A_N$…