Curve complexes versus Tits buildings: structures and applications

Tits buildings ∆Q(G) of linear algebraic groups G defined over the field of rational numbers Q have played an important role in understanding partial compactifications of symmetric spaces and compactifications of locally symmetric spaces, cohomological properties of arithmetic subgroups and S-arithmetic subgroups of G(Q). Curve complexes C(Sg,n) of surfaces Sg,n were introduced to parametrize boundary components of partial compactifications of Teichmüller spaces and were later applied to understand properties of mapping class groups of surfaces and the geometry and topology of 3-dimensional manifolds. Tits buildings are spherical building. Another important class of buildings consists of Euclidean buildings, for example, the BruhatTits buildings of linear algebraic groups defined over local fields. In this chapter, we summarize and compare some properties and applications of buildings and curve complexes. We try to emphasize their similarities but also point out differences. In some sense, curve complexes are combinations of spherical, Euclidean and hyperbolic buildings. We hope that such a comparison might motivate more questions and at the same time suggest methods to solve them. Furthermore it might introduce buildings to people who study curve complexes and curve complexes to people who study buildings. 2000 Mathematics Subject Classification: 53C35, 30F60, 22E40, 20G15, 57M99.