Curve complexes versus Tits buildings: structures and applications

Abstract

Tits buildings ${\Delta }_{\mathbb{Q}}(\mathbf{G})$ of linear algebraic groups $\mathbf{G}$ defined 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 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 Bruhat