**MSC:**- 20G25 Linear algebraic groups over local fields and their integers
- 20E42 Groups with a $BN$-pair; buildings, See also {51E24}

of $S$-arithmetic spin groups over number fields. After an exposition of

the relevant prerequesites from group cohomology, the geometry of the

Bruhat-Tits-building is used to get a concrete formula for the

Euler-Poincar\'e characteristic of an $S$-arithmetic spin group associated

to an positive definite quadratic space. Some examples illustrate the

connection between the geometry of the Bruhat-Tits-building and the

classification of quadratic lattices over a ring of integers.