A tale of two groups: arithmetic groups and mapping class groups

Abstract

In this chapter, we discuss similarities, differences and interaction between two natural and important classes of groups: arithmetic subgroups $\Gamma$ of Lie groups $G$ and mapping class groups mod of surfaces of genus $g$ with $n$ punctures. We also mention similar properties and problems for related groups such as outer automorphism groups $\mathrm{Out}(F_n)$, Coxeter groups and hyperbolic groups. Since groups are often effectively studied by suitable spaces on which they act, we also discuss related properties of actions of arithmetic groups on symmetric spaces and actions of mapping class groups on Teichmüller spaces.

Interaction between locally symmetric spaces and moduli spaces of Riemann surfaces through the example of the Jacobian map will also be discussed in the last part of this chapter. Since reduction theory, i.e., finding good fundamental domains for proper actions of discrete groups, is crucial to transformation group theory, i.e., to understand the algebraic structures of groups, properties of group actions and geometry, topology and compactifications of the quotient spaces, we discuss many different approaches to reduction theory of arithmetic groups acting on symmetric spaces. These results for arithmetic groups motivate some results on fundamental domains for the action of mapping class groups on Teichmüller spaces. For example, the Minkowski reduction theory of quadratic forms is generalized to the action of ${\mathrm{Mod}}_g={\mathrm{Mod}}_{g,0}$ on the Teichmüller space ${\mathcal{T}}_g$ to construct an intrinsic fundamental domain consisting of finitely many cells, solving a weaker version of a folklore conjecture in the theory of Teichmüller spaces.