On the fields generated by the lengths of closed geodesics in locally symmetric spaces

Abstract

i) If d 1 > d 2 then conditions (T 1 ) and (N 1 ) hold. (ii) If d 1 = d 2 but K Γ 1 ⊂ K Γ 2 then again conditions (T 1 ) and (N 1 ) hold. Thus, unless d 1 = d 2 and K Γ 1 = K Γ 2 , conditions (T i ) and (N i ) hold for at least one i ∈ {1, 2}.Assume now that d 1 = d 2 =: d and the subgroups Γ 1 and Γ 2 are arithmetic.(iii) If d is either even or is congruent to 3(mod 4), then either M 1 and M 2 are commensurable, hence length-commensurable and F 1 = F 2 , or (T i ) and (N i ) hold for at least one i ∈ {1, 2}.(iv) If d ≡ 1(mod 4) and in addition K Γ i = Q for at least one i ∈ {1, 2} then either M 1 and M 2 are length-commensurable (although not necessarily commensurable), or conditions (T i ) and (N i ) hold for at least one i ∈ {1, 2}.The results of [5] enable us to consider the situation where one of the groups is of type B n and the other is of type C n . the preparation of this paper, the second-named author was visiting the Mathematics Department of the University of Michigan as a Gehring Professor; the hospitality and generous support of this institution are thankfully acknowledged. Weak containmentThe goal of this section is to derive several consequences of the relation of weak containment (see Definition 1 of the Introduction) that will be needed later. We begin with some definitions and results for algebraic tori. Given a torus T defined over a field K, we let K T denote its (minimal) splitting field over K (contained in a fixed algebraic closure K of K). The following definition goes back to [9].Definition 4. A K-torus T is called K-irreducible (or, irreducible over K) if it does not contain any proper K-subtori.

The genus of a division algebra and the unramified Brauer group

“…If K is of characteristic zero, this is proved in [22,Corollary 3.2]; the argument in positive characteristic requires only minimal changes.…”

Division algebras with the same maximal subfields

“…Proposition 6.4 (cf. [37], Corollary 3.2). Let G be an absolutely almost simple algebraic group over a finitely generated field K .…”

Simple algebraic groups with the same maximal tori, weakly commensurable Zariski-dense subgroups, and good reduction

“…(cf. [55,Theorem 3.4]) Let G be a connected absolutely almost simple algebraic group over a finitely generated field k of characteristic zero, v be a discrete valuation of k such that the completion k v is locally compact, and T (v) be a maximal k v -torus of G. Given a finitely generated Zariski-dense subgroup Γ ⊂ G(k) whose closure in G(k v ) for the v-adic topology is open, there exists a regular semi-simple element γ ∈ Γ of infinite order such that the corresponding torus T = C G (γ) • is generic over k and is conjugate to T (v) by an element of G(k v ).…”