87 problems

Arad–Herzog conjecture. The product of two nontrivial conjugacy classes of $G$ is never a conjugacy class.

Let $R$ be a pro-$p$ ring, and let $G$ be a pro-$p$ group that is linear over $R$. Linearity conjecture. The group $G$ cannot contain a non-abelian free pro-$p$ subgroup. This exte…

Let $V$ be a finite vector space, let $G_0$ be an irreducible subgroup of $\operatorname{GL}(V)$, and let … For $g\in G_0$, write $C_V(g)$ for its fixed-point space. Neumann's affi…

Let $q$ be a prime power and let $G_n=\operatorname{GL}(n,q)$. Elements $x,y\in G_n$ are $t$-space-intersecting when there is a $t$-dimensional subspace $U\leq\mathbb{F}_q^n$ such…

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…

Breuillard's growth conjecture. For every integer $d>0$, there is $c_d>0$ such that

Let $\Gamma$ be a discrete linear group with Kazhdan's property (T). For each integer $n\geq 1$, let $\varkappa_n(\Gamma)$ denote the dimension-growth invariant used in the paper f…

Let $R={\mathbb Z}[t]$ or $R={\mathbb Z}[t,t^{-1}]$, and let $K(R)$ be the kernel of the evaluation map $SL_n(R)\to SL_n({\mathbb Z})$. Pseudo-nilpotence conjecture. The group…

Let $R={\mathbb Z}[t]$ or $R={\mathbb Z}[t,t^{-1}]$, and let $SL_n(R)$ be the corresponding special linear group. Cohomology vanishing conjecture. If $n\ge 3$, then … The paper giv…

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 $F$ be a finite field, let $n\ge 3$, and let … be the subgroup of matrices congruent to the identity modulo $t$. Write $H_1(K)$ for its first homology group and…

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…

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

Let $n$ be a positive integer and let $B$ tend to infinity. Choose a matrix from the set of matrices in $\operatorname{SL}(n,\mathbb{Z})$ whose coefficients have absolute value at…

Let $n$ and $r$ be integers with $r<n-1$, and let a continuous action of $SL(n,\mathbb{Z})$ be given on an $r$-dimensional mod 2 homology sphere. Mod 2 homology-sphere action conje…

Let $F_k$ be a free group of rank $k\ge 2$, let $\phi$ be an injective endomorphism of $F_k$, and let $\operatorname{HNN}_{\phi}(F_k)$ denote the corresponding ascending HNN extens…

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

Let $F$ be an infinite field and let $n$ be a positive integer. The natural homomorphism … is considered, where $H_n({\rm inc})$ denotes the map induced by the standard inclusion…

Let $R$ be a commutative pro-$p$ ring, let $G\subseteq GL_n(R)$ be a linear group over $R$, and let $\Gamma(p)$ be the profinite automorphism group of the rooted $p$-ary tree. The…

Let $R$ be a commutative Noetherian ring, let $m\geq 0$, and let $n\geq \max(3,\dim(R)+2)$. An elementary matrix over $R[x_1,\ldots,x_m]$ is a matrix differing from the identity ma…

Let $V$ be a finite vector space, let $G_0$ be an irreducible subgroup of $\operatorname{GL}(V)$, and suppose that the affine primitive group $G=V\rtimes G_0$ is solvable. For…

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 $\Gamma=\langle X\rangle$ be a finitely generated and sharply MIF group. For $w(x)\in \Gamma * \langle x\rangle$, write $\|w\|_{X\cup\{x,x^{-1}\}}$ for its word length with res…