29 problems

Let ${\bf w}$ be an $F$-root of ${\boldsymbol\pi}$, put $w=\beta({\bf w})$, and let $C_W(wF)$ be the centralizer of the corresponding regular element of $WF$. Let $B(w)$ be the bra…

Generalized centralizer theorem. For arbitrary $I\subseteq S$, if $\gamma\in\Pi^I$ and $s_\gamma\in W^{\perp I}{}_{\mathrm{fin}}$, then $w\cdot\gamma=\gamma$ for every $w\in Y_I$.…

Bessis–Digne–Michel's centralizer conjecture. The natural morphism

Uniform period bound conjecture. There is a number $n(G)$ such that $p\leq n(G)$ for every $e004\gamma i G$. This would provide a uniform bound for the centralizers needed by the ca…

Let $X$ be a compact homogeneous manifold, let $f_0\in\operatorname{Aff}(X)$ be a $K$-system, and suppose that $\mathcal{Z}(f_0)$ contains a hyperbolic affine diffeomorphism. If…

Strengthened row-maximum conjecture. For every $i$ with $1\leq i\leq \ell$,

Let $B$ be an Artin group with standard generating set ${\mathbf S}$, let ${\mathbf b}\in B$ have minimal length in its conjugacy class, and let $\supp({\mathbf b})$ denote its sup…

Centralizer-count and derived-order conjecture. If

Let $G$ be a Lie group, let $X=G/\Gamma$ be a homogeneous space with $\Gamma<G$ cocompact and discrete, and let $f_0=L_a\circ\Psi$ be an affine diffeomorphism, where $a\in G$ and…

Let $H_3(\mathbb R)$ be the three-dimensional Heisenberg group, let $\Gamma< H_3(\mathbb R)$ be a cocompact lattice, and let … Let $\phi_{\mathbb R}$ be a uniquely ergodic, or Diop…

Let $O_q$ be the $q$-Onsager algebra, and let $\tilde G_{k+1}$, $k\in\mathbb N$, be its alternating generators. For $x\in O_q$, consider the subalgebra generated by…

Centralizer-count conjecture. The number of distinct centralizers of $G$ satisfies

Shumyatsky's conjecture. If $C_G(g)$ has finite rank for every nontrivial $g\in G$, and $G$ is not a pro-$p$ group, then $G$ has finite rank.

Let $G$ and $H$ be finite simple groups. For a finite group $K$, let $2\text{-}\operatorname{Cent}(K)$ denote the set of centralizers of $2$-element subsets of $K$, and let…

Let $\mathfrak{g}$ be a semisimple Lie algebra of rank at least $3$, let $A\in\mathfrak{g}$ be nilpotent, and write $Z(A)$ for its centralizer and $\operatorname{rk}\mathfrak{g}$ f…

Finite-rank conjecture. Then $G$ has finite rank.

Centralizer–commutator conjecture. If

Let $M$ be a compact connected manifold, let $X$ be a $C^1$-generic vector field on $M$, and let $f:M\to\mathbb{R}$ be a function that is either $C^1$ or $C^0$. Call $f$ invariant…

Let $G$ be a finite group, let $Z(G)$ denote its center, and call $G$ induced regular when the induced non-centralizer graph on $G\setminus Z(G)$ is regular. A finite group is an e…

Let $G$ be a finite group, let $Z(G)$ be its center, and let $\Upsilon_G$ be the simple graph whose vertices are the elements of $G$, with distinct vertices $x$ and $y$ adjacent wh…

Let $m,n\geq 1$ be fixed integers, let $R$ be a semiprime ring satisfying suitable torsion restrictions, and let $T:R\longrightarrow R$ be a generalized $(m,n)$-Jordan centralizer.…

The Centralizer Conjecture over $D$. Suppose $A,B,w\in A_1(D)$ satisfy

The two-dimensional Centralizer Conjecture over $D$. Suppose

Let $G$ be a discrete group with finite asymptotic dimension, and for $g\in G$ let $C_g<G$ denote its centralizer. Centralizer quotient conjecture. The quotient … also has finite a…

Let $((S,\mathcal{F}),(T,\mathcal{E}))$ be a normal pair of fusion systems realized by a normal pair $(G,H)$ of finite groups. Write $O_{p'}(H)$ for the largest normal subgroup of…