Higher finiteness properties of reductive arithmetic groups in positive characteristic: The Rank Theorem

Abstract

We show that the finiteness length of an S-arithmetic subgroup Γ in a noncommutative isotropic absolutely almost simple group G over a global function field is one less than the sum of the local ranks of G taken over the places in S. This determines the finiteness properties for arithmetic subgroups in isotropic reductive groups, confirming the conjectured finiteness properties for this class of groups.Our main tool is Behr-Harder reduction theory which we recast in terms of the metric structure of euclidean buildings. for Mathematics (Bonn) for their hospitality within the trimester programs "Rigidity" and "Algebra and number theory". We also gratefully acknowledge the hospitality of the Mathematisches Forschungsinstitut Oberwolfach during several workshops and their RiP program. The DFG provided appreciated financial and structural support via SFB 701 at Bielefeld University and via GR 2077/5 and GR 2077/7 at Darmstadt.We are also indebted to Bernd Schulz for explaining his thesis. The participants of the Bielefeld seminar on reduction theory in summer 2010 (in particular Herbert Abels, Werner Hoffmann, Gregory Margulis, and Andrei Rapinchuk) provided valuable expertise. We thank

Semidualities from products of trees

“…A point of difference in the proof of our formulation of these results compared with formulations in other papers, is that we'll use the reduction theory from Bestvina-Eskin-Wortman [3] as an input, which has the advantage, though not directly applied in this paper, of being equally applicable to arithmetic groups defined with respect to a number field. See also Bux-Wortman [12] and Bux-Köhl-Witzel [10]. 4.1.…”

“…Note that Bux-Köhl-Witzel [10] shows that any Γ as in Conjecture 2 is of type F P k(G,S)−1 over Z [1/p], and it is well-known that cd Z[1/p] Γ ≤ k(G, S) since the dimension of X S equals k(G, S) (see Lemma 43 below). Therefore, proving Conjecture 2 would amount to proving that H m (Γ; Z[1/p]Γ) = 0 if m < k(G, S), and that D = H k(G,S) (Γ; Z[1/p]Γ) is flat as a Z[1/p]-module, the latter condition being equivalent to D being torsion-free since Z[1/p] is a principal ideal domain.…”

“…As an illustration, let L be a field whose characteristic is not equal to p. Recall that cd L (Γ) ≤ k(G, S). By Bux-Köhl-Witzel [10], L is of type F P k(G,S)−1 as a Z[1/p]Γ-module. Therefore if Conjecture 2 is true, then H 1 (Γ; L) is a quotient of H k(G,S)−1 (Γ; D), and if 2 ≤ n ≤ k(G, S) then H k(G,S)−n (Γ; D) ∼ = H n (Γ; L) Note that the only dimension of H * (Γ; L) which semiduality would not be able to help determine is dimension 0, but we know H 0 (Γ; L) = L.…”

Bounding the homological finiteness length

“…As in [11], to any K-group G, there is associated a non-negative integer k(G, S) = v∈S rank Kv G. [11]. In a recent remarkable paper, Bux, Gramlich and Witzel [8] showed that φ(H(O S )) = k(H, S) − 1. Calculating the homological finiteness length of non-uniform lattices on CAT(0) polyhedral complexes is an ambitious open problem.…”

Presentation of hyperbolic Kac–Moody groups over rings

“…Higher-dimensional finiteness properties are also very interesting. Their analysis for most S-arithmetic subgroups of algebraic groups has recently been completed by Bux, Köhl and Witzel [8]. One should be able to combine their results with corollary 3 to obtain higherdimensional finiteness properties in the setting of corollary 5.…”