Curve Complexes Versus Tits Buildings: Structures and Applications

Abstract

Tits buildings $${\Delta }_{\mathbb{Q}}(\mathbf{G})$$ of linear algebraic groupsGdefined over the field of rational numbers $$\mathbb{Q}$$ 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 $$\mathbf{G}(\mathbb{Q})$$ . Curve complexes $$\mathcal{C}({S}_{g,n})$$ of surfaces S g,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 geometry and topology of 3-dimensional manifolds. Tits buildings are spherical buildings, and another important class of buildings consists of Euclidean buildings, for example, the Bruhat–Tits buildings of linear algebraic groups defined over local fields. In this paper, 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 also suggest methods to solve them at the same time, and furthermore it might introduce buildings to people who study curve complexes and curve complexes to people who study buildings.