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
We determine when an arithmetic subgroup of a reductive group defined over a global function field is of type F P ∞ by comparing its large-scale geometry to the large-scale geometry of lattices in real semisimple Lie groups. * Supported by an NSF Postdoctoral Fellowship. 1 see e.g. the final introductory paragraph of [St 2]
The starting point of our investigation is the remarkable paper [2] in which Bestvina and Brady gave an example of an infinitely related group of type FP2. The result about right‐angled Artin groups behind their example is best interpreted by means of the Bieri–Strebel–Neumann–Renz Σ‐invariants.
For a group G the invariants Σn(G) and Σn(G, Z) are sets of non‐trivial homomorphisms χ:G→R. They contain full information about finiteness properties of subgroups of G with abelian factor groups. The main result of [2] determines for the canonical homomorphism χ, taking each generator of the right‐angled Artin group G to 1, the maximal n with χ ∈ Σn(G), respectively χ ∈ Σn(G, Z).
In [6] Meier, Meinert and VanWyk completed the picture by computing the full Σ‐invariants of right‐angled Artin groups using as well the result of Bestvina and Brady as algebraic techniques from Σ‐theory. Here we offer a new account of their result which is totally geometric. In fact, we return to the Bestvina–Brady construction and simplify their argument considerably by bringing a more general notion of links into play. At the end of the first section we re‐prove their main result. By re‐computing the full Σ‐invariants, we show in the second section that the simplification even adds some power to the method. The criterion we give provides new insight on the geometric nature of the ‘n‐domination’ condition employed in [6].
Let G be a Chevalley group scheme and B ≤ G a Borel subgroup scheme, both defined over Z. Let K be a global function field, S be a finite non-empty set of places over K , and O S be the corresponding S -arithmetic ring. Then, the Sarithmetic group B(O S ) is of type F |S|−1 but not of type FP |S| . Moreover one can derive lower and upper bounds for the geometric invariants Σ m (B(O S )). These are sharp if G has rank 1. For higher ranks, the estimates imply that normal subgroups of B(O S ) with abelian quotients, generically, satisfy strong finiteness conditions.
Abstract. We compute the VCD of the group of partially symmetric outer automorphisms of a free group. We use this to obtain new upper and lower bounds on the VCD of the outer automorphism group of a 2-dimensional right-angled Artin group. In the case of a right-angled Artin group with defining graph a tree, the bounds agree.
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