8 problems

Let $\mathbf F_n$ be the free lattice on $n$ generators. An elementary extension $\mathbf K\succ\mathbf F_n$ is standard if the associated map $h_{\mathbf K}$ has range in…

Let $\mathcal G(\Gamma,\{G_v\}_{v\in\Gamma})$ and $\mathcal G(\Gamma',\{G'_v\}_{v\in\Gamma'})$ belong to the specified class $\mathcal G(\mathbf F_N,\mathbf C^r_\Phi)$ with $N>0$.…

Let $\mathbb G(\Gamma)$ and $\mathbb G(\Delta)$ be right-angled Artin groups, and let $\operatorname{Core}(\Gamma)$ denote the core graph. RAAG core classification conjecture. If ……

Let $\mathcal G=\mathcal G(\Gamma,\{G_v\}_{v\in\Gamma})$ and $\mathcal G'=\mathcal G(\Delta,\{G'_v\}_{v\in\Delta})$ be graph products with finite graphs, where each vertex group sa…

Let graph products satisfy the hypotheses of the preceding elementary-equivalence theorem. Reducedness necessity conjecture. The conditions on the graphs—namely, that they be almos…

Let $G=\Gamma(\Gamma,\{\mathbb Z\})$ and $G'=\Gamma(\Gamma',\{\mathbb Z\})$ be right-angled Artin groups. Core conjecture. Then … if and only if the cores of $G$ and $G'$ are isomo…

Let $Q$ be a finite group, let $F_{1}$ and $F_{2}$ be non-abelian free groups, and for $i=1,2$ let $\pi_i:F_i\twoheadrightarrow Q$ be epimorphisms. Let $\mathcal{L}^{Q}$ be the lan…

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…