The cubical route to understanding Artin groups

Objective

Artin groups occupy an important place in modern mathematics, sitting at the intersection of algebra, geometry, and topology. They generalise free groups, free abelian groups, and braid groups - already central objects with deep connections to knot theory, low-dimensional topology, and mathematical physics - while extending this richness across a vast family of groups associated with Coxeter systems. Despite their introduction in the 1960s, and the extensive body of work that has arisen from their study, many of the most basic questions about Artin groups remain unresolved. The goal of this proposal is to study Artin groups as quotients of cubulated groups by developing and exploiting the tools of cubical small-cancellation theory, with the aim of tackling central open questions, such as the K(pi,1) conjecture and residual finiteness, and foundational algorithmic challenges, such as the word and conjugacy problems.