44 problems

Let $F_n$ be the free group of rank $n$, let $\mathrm{IA}_n=\mathcal{A}_n'(1)$ be the group of automorphisms of $F_n$ acting trivially on its abelianization, and let … be the lower…

Let $G$ be a residually nilpotent group. Write $H_1(G)$ and $H_2(G)$ for its integral first and second homology groups, and let $\operatorname{Hom}(H_2(G),\mathbb Z)$ denote the gr…

Let $G$ be the group considered in the paper, let $H_i$ be the associated upper-triangular subgroups, and let $\lambda_i(G)$ and $\gamma_i(G)$ denote the terms of the lower exponen…

Ellis's conjecture. If $P$ is a free group, then $\theta_n$ are isomorphisms for all $n\geq 1$. This is the precise multiplicative-Lie-algebra formulation of the preceding universa…

Let $A_n$ be the free associative algebra on $n$ generators, regarded as a Lie algebra, and consider the surjective map … The relations in its kernel include the Jacobi relations,…

Let $A_n$ be the free associative algebra on $n$ generators, regarded as a Lie algebra, and let $A_{n,k}$ denote its $k$-th consecutive quotient of the lower central series. Write…

Let $\Gamma_n$ be the pure cactus group, let $\Gamma_n^k$ denote its lower central series, and let $\mathcal{L}_n$ be the associated graded Lie algebra after quotienting by $2$-tor…

Graphic LCS formula.

MLS LCS formula. If ${\mathcal A}$ is an MLS arrangement with group $G$, then, for all $k\ge 2$,

Let $\mathcal{A}$ be a complex hyperplane arrangement with arrangement group $G=G(\mathcal{A})$. Let $R_1(\mathcal{A})=\bigcup_i L_i$ be its resonance variety, and set…

Let $\mathcal{K}$ be a flag complex, let $L_\mathcal{K}$ be the graph Lie algebra over $\mathbb{Z}_2$, and let $N_\mathcal{K}$ be the Lie subalgebra generated by the Grbić–Panov–Th…

Zelmanov's conjecture. Every just infinite pro-$p$ group of finite lower central width is either soluble, $p$-adic analytic, or commensurable to a positive part of a loop group or…

Let $G$ be a generalized Fabrykowski–Gupta group, and let $p$ be the prime associated with the group. Write $\gamma_i(G)$ for the $i$th term of the lower central series of $G$. Bar…

Dimension subgroup conjecture. The dimension subgroup series with respect to the integers coincides with the lower central series:

Experimental stopping-index conjecture. The lower central series of ${\mathbf B}_{1,m}({\mathbb P}^2)$ always stops at

Projective-plane braid stopping conjecture. The lower central series of ${\mathbf B}_{1,m}({\mathbb P}^2)$ stops at

Parity conjecture. The value is always $\epsilon=1$ when $m$ is even and $\epsilon=0$ when $m$ is odd; equivalently,

Freedman–Hain–Teichner conjecture. There is no nilpotent group $G$ satisfying this isomorphism and

Let $L=(L_1,\dots,L_n)$ be an $n$-component link in $S^3$, let $G=\pi_1(S^3\setminus\bigcup_{i=1}^n L_i)$ be its link group, and let $F_{n-1}$ be the free group of rank $n-1$. For…

Malestein–Putman conjecture. The lengths of these shortest non-trivial elements have asymptotic behaviour of the form

Let $A_n$ be the free associative algebra on $n$ generators, and let $j=(2,\dots,2)\in\mathbb{N}^k$. All-two tuple conjecture. … The formula is motivated by combinatorial considera…

Let $A_3$ be the free associative algebra on three generators. For a tuple $i=(i_1,\dots,i_k)$ with $i_p\geq 2$ for every $p$, let $\mathrm{I}(A_3,i)$ denote the maximal lower-cent…

Even-index product conjecture. If $i,j\geq 2$ are both even, then in general

Let $k\geq 1$ and let $\gamma_k=\gamma_k(x_1,\dots,x_k)$ be the lower central word, with $\gamma_k(G)$ denoting the closed subgroup generated by all $\gamma_k$-values in a profinit…

Let $A=\mathbb{Z}\langle x,y\rangle/(x^m+y^m)$, and let $N_3(A)$ denote the third lower central series quotient, with $N_3(A)[d]$ its degree-$d$ component. Rank formula conjecture.…