4 problems

Let $\Lambda$ and $\Lambda'$ satisfy the assumptions of the source's six-cycle link lemma, with $\Lambda'$ not necessarily admissible in $\Lambda$. Consider the links of vertices i…

Let $A_S$ be an Artin group, and let $S'ubset S$ be almost spherical. Let $\Delta_{S,S'}$ denote the relative Artin complex spanned by vertices of types indexed by $S'$. Assume tha…

Artin groups' algebraic and $K(\pi,1)$ conjectures. For every Artin group $A_{\Gamma}$: (1) $A_{\Gamma}$ has solvable word and conjugacy problems; (2) $A_{\Gamma}$ is torsion-free;…

Let $L$ be the nerve of an Artin group $A_L$, and let $\mathcal{S}(L)$ be the poset of simplices of $L$ together with the empty simplex. The poset of groups over $\mathcal{S}(L)$ h…