Olga Varghese

Automatic continuity for Artin groups

The conjecture we address says that any abstract group homomorphism from a locally compact Hausdorff group into an Artin group is continuous, i. e. all Artin groups are lcH-slender. We use the clique-cube complex $C_\Gamma$ associated to the Artin group $A_\Gamma$ to reduce the automatic continuity conjecture to Artin groups where the defining graphs are complete. Under mild algebraic conditions on small parabolic subgroups of $A_\Gamma$ we show that if all special complete subgroups $A_\Delta$ of $A_\Gamma$ are lcH-slender, then $A_\Gamma$ is lcH-slender.