Cube complexes, particularly those that satisfy a form of non-positive curvature called the CAT(0) condition, have played a major role in recent work in geometry and low-dimensional topology. They were a vital tool in the celebrated work of Agol and Wise proving Thurston's virtual fibring conjecture. They feature prominently in the study of decompositions of groups into graphs of groups. Combinatorially, they are the realisation spaces of discrete median algebras, and as such have been studied from the point of view both of combinatorics and of lattice theory. They have also found applications to understanding the coarse geometry of the mapping class group and Teichmüller spaces, which have recently been shown to have a natural coarsely cubical structure.
There is considerable potential for interaction between researchers who have studied such structures from differing perspectives, and who have diverse insights into the subject. The purpose of this workshop is to mine this potential to solve old problems, such as whether CAT(0) cube complexes with cocompact actions can contain periodic flats, and to develop it in new directions. These include studying other combinatorial structures which arise in low-dimensional topology such as the arc and curve complexes associated to a surface, where the combinatorial notion of ''dismantlability," borrowed from graph theory has begun to be used profitably. There is also direct interaction with the study of outer automorphism groups, via spaces made of CAT(0) cube complexes with fundamental group a given right-angled Artin group, and via complexes such as the free splitting and free factor complexes.
For further information please visit https://www2.warwick.ac.uk/fac/sci/maths/research/events/2017-18/symposium/gscc/.