On Garland's Vanishing Cohomology Theorem - a study of the Bruhat-Tits building of SL(V )

Specialeforsvar ved Karen Elise Løvengreen

Titel: On Garland's Vanishing Cohomology Theorem - a study of the Bruhat-Tits building of SL(V )

 Abstract: This thesis exposes Garland's combinatorial proof of vanishing simplical cohomology. Abstract simplicial complexes and chamber complexes are briefly studied in order to interpretatively define the notion of a building. The main example of a building will be the Bruhat-Tits building of $\SL_n(K)$, $\mB$, with $K$ a non-Archimedian local field. The simplicial cochain groups are equipped with an inner product with respect to which the coboundary map $d$ has an adjoint $\delta$. It is studied how the cochain group endomorphism, $\Delta=\delta d$, acts locally, namely within the link of a vertex. With $\Gamma$ a cocompact subgroup of $\SL_n(K)$ preserving the link structure of vertices of $\mB$, it is proved that the simplicial cohomology groups for the orbit space $\mB/\Gamma$ vanish if the minimal non-zero eigenvalues of the local versions of $\Delta$ are sufficiently large. The combinatorially flavoured approach makes the proof method intriguing in itself.

 

Vejleder: Henrik Schlichtkrull
Censor:   Martin Raussen, Aalborg Universitet