18 problems

Let $n\leq m$ be positive integers, and let $H$ be a subgroup of the direct product … where each $G_i$ is of type $F_n$ for $1\leq i\leq m$. Say that $H$ is virtually surjective on…

Direct-product conjecture. $S$ has a subgroup of finite index that is itself a direct product of $n$ or fewer limit groups.

Direct-product conjecture. The clean graphs satisfy

Countable subdirect product conjecture. The answer to the first question is affirmative, while the answer to the second question is negative: there exist infinite non-periodic semi…

Bridson's conjecture. Direct products of free groups are profinitely rigid among finitely presented, residually finite groups.

Let $K_n$ be a complete graph with $n\geq 3$, and let $G$ be a nontrivial graph. Write $\kappa(G)$ for the connectivity of $G$ and $\delta(G)$ for its minimum degree. The direct pr…

Let two planar flows each have an attracting homoclinic saddle loop, and consider their direct product. The loops are the attracting homoclinic saddle loops of the respective facto…

Let $G \times H$ be a group, let $(u,v) \in G \times H$, and let $(u,v)^{G \times H}$ denote the normal closure of $(u,v)$ in $G \times H$. The associated subgroups…

Let $H_1,\ldots,H_n$ be complete multipartite graphs, and let $G=\prod_{i=1}^n H_i$. The independence number $\alpha(G)$ is the largest size of an independent set, and…

Let $G=\prod_{i=1}^n K[u_i,t_i]$ be a direct product of balanced complete multipartite graphs. The independence number $\alpha(G)$ is the largest size of an independent set, and…

Finite-subset covering conjecture. There exist sets $X_1,\dots,X_l\subseteq X$ whose union is $X$, such that for every $i$, either $X_i$ is contained in a hyperplane of the form…

Let $G$ be a finite group, let $G^n$ denote its $n$-th direct power, and let $A$ be a generating set of minimum size for $G^n$. Write ${\mathrm{rank}}(G)$ for the minimum size of a…

Let $G$ be a finite group, let $G^n$ denote its $n$-th direct power, and let $D(G^n)$ denote the diameter with respect to a generating set, using words in that set. Write…

Let ${\mathbf{A}}$ and ${\mathbf{B}}$ be expanded groups such that every congruence of ${\mathbf{A}}\times{\mathbf{B}}$ is a product congruence. Let $f$ and $g$ be unary polynomial…

Let $G$ and $H$ be groups, and let $k$ be an integer with $k\geq 2$. Write $\times^{k}G$ and $\times^{k}H$ for the direct products of $k$ copies of $G$ and $H$, respectively. Manev…

Strong product conjecture. Then $S\times T$ is strongly Markov.

Let $k\geq 2$, let $\Gamma_1,\dots,\Gamma_n$ be groups of type $\operatorname{FP}_k$, and let $P\leq\Gamma_1\times\dots\times\Gamma_n$ be a subgroup of their direct product. Say th…

Let $k\geq 2$. Let $\Gamma_1,\dots,\Gamma_n$ be groups of type $\operatorname{F}_k$, and let $P\leq\Gamma_1\times\dots\times\Gamma_n$ be a subgroup of their direct product. Say tha…