51 problems

Let $G$ be a finitely generated group, and define its Schreier girth by … where $S$ is a minimal generating set and $S\sim_{\mathcal{S}}S'$ means that $S'$ can be obtained from $S$…

Let $G\subseteq GL(n,\mathbb{C})$ be a finitely generated subgroup. An element of $G$ is primitive if it belongs to a generating set obtainable from the fixed generating set by fin…

Finite generation conjecture. The group $\Gamma$ is never finitely generated.

Conjecture. The groups $B$ and $P$ are linear and satisfy all the properties enumerated in that theorem whenever $W$ is an irreducible complex reflection group.

Let $R={\mathbb Z}[t]$ or $R={\mathbb Z}[t,t^{-1}]$, let $K(R)$ be the kernel of the evaluation map $SL_2(R)\to SL_2({\mathbb Z})$, and let ${\mathcal W}$ be the kernel of the indu…

Let $k$ be the field appearing in the power-series congruence subgroup, let $n,m$ be positive integers, let $p$ be a prime, and write $C_{n,m}(k[[t_1,t_2,\dots,t_m]])$ for the corr…

Exactness conjecture. This sequence is exact.

Bounded-generation conjecture. If $\Gamma$ is any noncocompact lattice in either

Gersten's conjecture. There is a recursive function $f\colon\mathbb{N}\to\mathbb{N}$, such that every finite presentation of a linear group has

Non-linearity conjecture. If $W$ is Gromov-hyperbolic and $q>\!\!>1$, then $\Lambda$ is not linear.

Let $p\in\mathbb{N}$ be prime, let $q$ be a power of $p$, and let $\mathrm{GL}_{n,q}$ denote the general linear group over the field with $q$ elements. A family is $p$-singular if…

Let $\mathbb F$ be an algebraically closed field, let $d\geq1$, and let $A\subset\operatorname{GL}_d(\mathbb F)$ be a $K$-approximate group. Suppose every $a\in A^2$ has an eigensp…

Let $\mathbb F$ be a field, let $d\geq1$, and let $A\subset\operatorname{GL}_d(\mathbb F)$ be a $K$-approximate group for some $K\geq1$. Polynomial-dimension abelian-structure conj…

Let $R$ be a commutative pro-$p$ ring and let $G\le \mathrm{GL}_n(R)$ be a linear group over $R$. The group $W_p$ is the group of $p$-adic automorphisms, namely a Sylow pro-$p$ sub…

General counting conjecture. If $\gcd(k,w+k+1)=1$, then the number of tame $\operatorname{SL}_k$-frieze patterns of width $w$ over $K$ is

Platonov–Potapchik's conjecture. If all primitive elements of $G$ are unipotent, then the group $G$ generated by $S$ is unipotent.

PSL2 congruence conjecture.

Characteristic-two PSL2 conjecture.

Salehi-Golsefidy–Varjú conjecture. A finitely generated subgroup $G<\operatorname{SL}_n(\mathbb Z)$ has super approximation with respect to all positive integers exactly when the i…

Lyndon–Ullman–Kim–Koberda conjecture. For every nonzero rational value of $q$ in $(-4,4)$, the group $\Delta_q$ is not free.

For $n\in\mathbb{N}$, define … Here $\mathscr{O}_K$ denotes the ring of integers of the number field $K$, and $\Delta_k$ is the strong boundedness invariant considered in the paper…

Let $q$ be a prime power and let $G_n=\operatorname{GL}(n,q)$. Let $Y,Z\subseteq G_n$ be $t$-space-cross-intersecting sets, meaning that every pair in $Y\times Z$ is $t$-space-inte…

Let $q$ be a prime power and let $G_n=\operatorname{GL}(n,q)$. A subset $Y\subseteq G_n$ is $t$-space-intersecting when every pair of elements of $Y$ is $t$-space-intersecting. For…

Let $q$ be a prime power and let $G_n=\operatorname{GL}(n,q)$. For positive integers $t$ and sufficiently large $n$ compared to $t$, let $Y\subseteq G_n$ be a $t$-intersecting set,…

Let $K$ be a finite field with $|K|=q$, let $\mathbb{F}$ be a free group, and let $w\in\mathbb{F}$. Let $\mathcal{R}$ be the algebra of character-like functions on the groups…