Finiteness properties of arithmetic groups over function fields

Abstract

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]

Presentations of Inverse Semigroups, Their Kernels and Extensions

“…This method admits a natural and straightforward generalisation to inverse semigroups and inverse semigroup presentations; see [17] for details. It is a consequence of these general results that, with the notation above, the following presentation, which we denote by P, [15] Presentations of kernels and extensions 303 defines the kernel K: the generating set is B, and the relations are…”

A geometric proof that SL2(ℤ[t,t−1]) is not finitely presented

“…It seems likely that more results along these lines can be proved, but it is not clear to us how much the results in [4] can be generalized. Below we phrase a question that seems a good place to start.…”

“…It is essentially a special case of our proof that arithmetic subgroups of SL n over global function fields are not of type FP 1 [4]. The proof uses a result of K. Brown's which requires the action to have "nice" stabilizers.…”

“…In this paper we stripped down the general proof of the main result from [4] to the special case of showing that SL 2 ‫ކ‬ q OEt; t 1 is not of type FP 2 , and then made some modest alterations until we arrived at the proof of Proposition A presented below.…”

“…We wrote this paper to show how the techniques in [4] might be applied to a more general class of groups.…”