A tale of two groups: arithmetic groups and mapping class groups
Handbook of Teichmüller Theory, Volume III
In this chapter, we discuss similarities, differences and interaction between two natural and important classes of groups: arithmetic subgroups Γ of Lie groups G and mapping class groups Mod g,n of surfaces of genus g with n punctures. We also mention similar properties and problems for related groups such as outer automorphism groups 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
... ss related properties of actions of arithmetic groups on symmetric spaces and actions of mapping class groups on Teichmüller spaces, hoping to get across the point that it is the existence of actions on good spaces that makes the groups interesting and special, and it is also the presence of large group actions that also makes the spaces interesting. 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 Mod g = Mod g,0 on the Teichmüller space T g to construct an intrinsic fundamental domain consisting of finitely many cells, * Partially Supported by NSF grant DMS 0905283 2 solving a weaker version of a folklore conjecture in the theory of Teichmüller spaces. On several aspects, more results are known for arithmetic groups, and we hope that discussion of results for arithmetic groups will suggest corresponding results for mapping class groups and hence increase interactions between the two classes of groups. In fact, in writing this survey and following the philosophy of this chapter, we noticed the natural procedure in §5.12 of constructing the Deligne-Mumford compactification of the moduli space of Riemann surfaces from the Bers compactification of the Teichmüller space by applying the general procedure of Satake compactifications of locally symmetric spaces, i.e., how to pass from a compactification of a symmetric space to a compactification of a locally symmetric space by making use of the reduction theory for arithmetic groups. The layout of this chapter is as follows. In §1, the introduction, we discuss some general questions about discrete groups, group actions and transformation group theories. In §2, we summarize results on arithmetic subgroups Γ of semisimple Lie groups G and mapping class groups Mod g,n of surfaces of genus g with n punctures, and their actions on symmetric spaces of noncompact type and Teichmüller spaces respectively. 1 For comparison and for the sake of completeness, we also discuss corresponding properties of three related classes of groups: outer automorphism groups of free groups, Coxeter groups and hyperbolic groups. In §3, we describe several sources where discrete groups and discrete transformation groups arise. In §4 and §5 we give definitions and details of some of the properties listed earlier in §2 for arithmetic groups and mapping class groups. In the last section, §6, we deal with the coarse Schottky problem, a large scale geometric generalization of the classical Schottky problem of characterizing the Jacobian varieties among abelian varieties, i.e., the image of the Jacobian map from the moduli space M g of compact Riemann surfaces of genus g to the Siegel modular variety A g , an important Hermitian locally symmetric space, which is equal to the moduli space of principally polarized abelian varieties of dimension g. For the detailed organization of this chapter, see the table of contents starting on the next page.