Infinite topology of curve complexes and non-Poincaré duality of Teichmüller modular groups

Let S be an orientable surface. Let Diff(S) be the group of all diffeomorphisms of S, and Diff(S) its identity component. Then Mod±S = Diff(S)/Diff (S) is called the extended mapping class group or the extended Teichmüller modular group of S. Let Diff(S) be the subgroup of orientation preserving diffeomorphisms of S. Then Mod(S) = Diff(S)/Diff(S) is called the mapping class group or the Teichmüller modular group of S. When S is a closed surface of genus 1, for example, when S = Z\R is the standard torus, then Mod(S) can be identified with GL(2,Z), and Mod(S) can be identified with SL(2,Z). If S is a closed surface of genus g ≥ 2, or more generally an oriented surface of negative Euler characteristic χ(S), then Mod(S) and Mod(S) can be considered as natural generalizations of GL(2,Z) and SL(2,Z) respectively. The group SL(2,Z) is an important example of arithmetic subgroups of semisimple linear algebraic groups, in particular, it is the first group in the classical family of arithmetic subgroups SL(n,Z), n ≥ 2 of SL(n,R). Arithmetic subgroups Γ of semisimple linear algebraic groups G (defined over Q) enjoy many good properties and are special among discrete groups. For example, they are finitely presented and enjoy other finiteness properties such as being of type FP∞ and of type FL. Borel and Serre showed in [BoS] that arithmetic subgroups Γ are virtual duality groups and their virtual cohomological dimension can be computed explicitly (we outline parts of this theory below). The cohomology groups of a natural family of arithmetic subgroups such as SL(n,Z) stabilize as n → ∞. See [Se] and [Bo1] for a summary and [Bo2] for a computation of the stable real cohomology groups. Let G = G(R) be the real locus of G, let K ⊂ G be a maximal compact subgroup, and let X = G/K be the associated symmetric space. Assume that Γ is a torsion-free arithmetic subgroup of G(Q). Then the locally symmetric space Γ\X is an aspherical manifold with the fundamental group Γ. If Γ is a cocompact subgroup of G, then the Poincaré duality for Γ\X implies that Γ is a Poincaré duality group. On the other hand, if the arithmetic subgroup Γ is not a cocompact discrete subgroup of G (as is SL(n,Z), for example), then [BoS] (see the notes by Serre near the end [BiE]) implies that Γ is not a Poincaré duality group, in particular, it can not be realized as the fundamental group of a closed aspherical manifold. Instead of being a Poincaré duality group, Γ enjoys a weaker property being a duality group in the sense of Bieri-Eckmann [BiE]. In Section 2, we will recall the definition of duality groups and Poincaré duality groups (see, in particular, the formulas (1), (2), 3)̇ . In the context of Teichmüller modular groups, the Teichmüller space TS of S plays the role of the symmetric space X, and the canonical action of ModS on TS plays the role of the action of Γ on X. This analogy was discovered by Harvey [Harv1], [Harv2] who, in particular, was motivated by the problem of providing analogues of some constructions of Borel-Serre [BoS].