Stable Real Cohomology of Arithmetic Groups II

Abstract

Given a discrete subgroup Γ of a connected real semisimple Lie group G with finite center there is a natural homomorphism

$$j_\Gamma ^q:I_G^q \to {H^q}\left( {\Gamma ;c} \right)\quad \left( {q = 0,1, \ldots } \right),$$

((1))

where I ^q_G denotes the space of G-invariant harmonic q-forms on the symmetric space quotient X=G/K of G by a maximal compact subgroup K. If Γ is cocompact, this homomorphism is injective in all dimensions and the main objective of Matsushima in [19] is to give a range m(G), independent of Γ, in which j ^q_Γ is also surjective. The main argument there is to show that if a certain quadratic form depending on q is positive non-degenerate, then any Γ-invariant harmonic q-form is automatically G-invariant. In [3], we proved similarly the existence of a range in which j ^q_Γ is bijective when Γ is arithmetic, but not necessarily cocompact. There are three main steps to the proof: (i) The cohomology of Γ can be computed by using differential forms which satisfy a certain growth condition, “logarithmic growth,” at infinity; (ii) up to some range c(G), these forms are all square integrable; and (iii) use the fact, pointed out in [16] , that for q ≦ m(G), Matsushima’s arguments remain valid in the non-compact case for square integrable forms.