A canonical arithmetic quotient for actions of lattices in by David Fisher

By David Fisher

During this paper we produce an invariant for any ergodic, finite entropy motion of a lattice in an easy Lie workforce on a finite degree house. The invariant is basically an equivalence classification of measurable quotients of a definite sort. The quotients are basically double coset areas and are created from a Lie team, a compact subgroup of the Lie crew, and a commensurability type of lattices within the Lie team.

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In particular, (a) and (b) hold whenever (c) A and A are stably associated, and in a weakly finite ring (a)–(c) are equivalent for two matrices of the same index. Proof. Suppose that A, A satisfy a comaximal relation (7), say AD − BC = I, D A − C B = I. (8) Then on writing M= A C B D and N= D −C −B A , (9) Generalities on rings and modules 32 I P we have M N = 0 , where P = C D − DC . Hence M has the right I inverse D −C −B A I −P 0 = I ∗ ∗ −B A , and (b) follows. Conversely, if N in (9) is a right inverse of M, then (7) and (8) hold, hence (7) is then a comaximal relation.

If α : Q → P is stably associated to α : Q → P , then α ⊕ 1 P is associated to 1 P ⊕ α . Hence two matrices A ∈ r R m and A ∈ sR n are stably associated, qua maps, if and only if A ⊕ In is associated to Im ⊕ A . In terms of matrices we obtain the following criteria by taking P, P , Q, Q to be free. 5. Let A ∈ r R m , A ∈ sR n be any two matrices. Then of the following, (a) and (b) are equivalent and imply (c): (a) there exists an (r + n) × (s + m) matrix ∗ A ∗ ∗ ∗ with an inverse of the ∗ , A (b) A and A are stably associated, (c) A and A are left similar.

Given a field extension E/k, write R E = R ⊗k E and for any R-module M denote the extension M ⊗k E by M E . If M, N are R-modules such that Hom R (M, N ) is finite-dimensional over k, then Hom R (M, N ) ⊗k E ∼ = Hom R E (M E , N E ). 6 Eigenrings and centralizers 35 Proof. There is a natural map from the left- to the right-hand side in (4), which is clearly injective, so it will be enough to show that both sides have the same dimension. Let (ei ), ( f λ ) be bases for M, N as k-spaces (possibly infinite-dimensional); then the action of R is given by ei x = ρi j (x)e j , fλ x = σλμ (x) f μ (x ∈ R), and Hom R (M, N ) is the space of all solutions α over k of the system ρi j (x)α jμ = j αiλ σλμ (x).

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