Finiteness Properties of Arithmetic Groups Acting on Twin Buildings by Stefan Witzel

Author:Stefan Witzel
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

It is a common situation to have a group G that acts on a polyhedral complex X with the properties that X is contractible and the stabilizers of cells are finite but X is not compact modulo the action of G. One is then interested in a G-invariant subspace X 0 of X that is compact modulo G and still has some desirable properties, in our case to be highly connected.

A useful technique to produce such a subspace is combinatorial Morse-theory which was developed by Bestvina and Brady. To apply it, one has to construct a G-invariant Morse-function whose sublevel sets are G-cocompact and whose descending links are highly connected. The Morse lemma then shows that the sublevel sets are highly connected and one can take X 0 to be one of them. Of course there has to remain some work to be done, which is to construct an appropriate Morse-function and analyze the descending links. This is what we will do in this chapter. But first we translate our algebraically described problem into this geometric setting.

Let G be a connected, non-commutative, absolutely almost simple -group. In this chapter we determine the finiteness length of where G is a connected, non-commutative, absolutely almost simple -group. We have seen in Sect. 1.​9 that acts strongly transitively on a locally finite irreducible Euclidean twin building and we will see that is the stabilizer in of a point of the twin building. Our goal is therefore to prove:

Theorem 2.1.

Let be an irreducible, thick, locally finite Euclidean twin building of dimension n. Let E be a group that acts strongly transitively on and assume that the kernel of the action is finite. Let be a point and let be the stabilizer of a − . Then G is of type F n−1 but not of type Fn .

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