Connectivity properties of horospheres in Euclidean buildings and applications to finiteness properties of discrete groups

Abstract

Let G(O S ) be an S-arithmetic subgroup of a connected, absolutely almost simple linear algebraic group G over a global function field K. We show that the sum of local ranks of G determines the homological finiteness properties of G(O S ) provided the K-rank of G is 1. This shows that the general upper bound for the finiteness length of G(O S ) established in an earlier paper is sharp in this case.The geometric analysis underlying our result determines the conectivity properties of horospheres in thick Euclidean buildings.

Lipschitz connectivity and filling invariants in solvable groups and buildings

“…Theorem 1.7 is similar to Theorem 7.7 of [BW11], and gives a higher-order version of Theorem 1.1 of [Dru04] for buildings and products of buildings. (Though note that Theorem 1.1 of [Dru04] applies to R-buildings as well as discrete buildings.…”

“…We will also apply the methods of Theorem 1.3 to horospheres in euclidean buildings and to the S-arithmetic groups considered in [BW11].…”

Semidualities from products of trees

“…We shall call a tuple (r Q ) Q∈P ∈ (R ∪ {∞}) P sufficiently large if the resulting sets B Q,S,r Q are pairwise disjoint, and if their pairwise distance is bounded below by a constant that is sufficiently large. It's known that if (r Q ) Q∈P is sufficiently large then X S,(r Q ) is (|S| − 2)connected but not (|S| − 1)-connected (see Stuhler [17], Bux-Wortman [12], and Bux-Köhl-Witzel [10]), but these topological properties are not directly relevant to this paper. What we require in this paper, and what we will prove in this section, is that H k c (X S,(r Q ) ) = 0 if k ≤ |S| −1 and (r Q ) Q∈P is sufficiently large.…”

“…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.…”

An infinitely generated virtual cohomology group for noncocompact arithmetic groups over function fields

“…Important evidence for the theorem of Bux-Köhl-Witzel was contributed by Behr [3], Abels [1], Abramenko [2], and Bux-Wortman [7].…”