10 problems

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

Multiplicativity conjecture. For any two such groups,

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 $k\geq 2$, 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…