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]

Abstract. We provide partial results towards a conjectural generalization of a theorem of Lubotzky-Mozes-Raghunathan for arithmetic groups (over number fields or function fields) that implies, in low dimensions, both polynomial isoperimetric inequalities and finiteness properties.As a tool in our proof, we establish polynomial isoperimetric inequalities and finiteness properties for certain solvable groups that appear as subgroups of parabolic groups in semisimple groups, thus generalizing a theorem of Bux.We also develop a precise version of reduction theory for arithmetic groups whose proof is, for the most part, independent of whether the underlying global field is a number field or a function field.

The dilation of a Euclidean graph is defined as the ratio of distance in the graph divided by distance in R d . In this paper we consider the problem of positioning the root of a star such that the dilation of the resulting star is minimal. We present a deterministic O(n log n)-time algorithm for evaluating the dilation of a given star; a randomized O(n log n) expected-time algorithm for finding an optimal center in R d ; and for the case d = 2, a randomized O(n2 α(n) log 2 n) expected-time algorithm for finding an optimal center among the input points.

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.

The dilation of a Euclidean graph is defined as the ratio of distance in the graph divided by distance in R d . In this paper we consider the problem of positioning the root of a star such that the dilation of the resulting star is minimal. We present a deterministic O(n log n)-time algorithm for evaluating the dilation of a given star; a randomized O(n log n) expected-time algorithm for finding an optimal center in R d ; and for the case d = 2, a randomized O(n 2 α(n) log 2 n) expected-time algorithm for finding an optimal center among the input points.