Discrete Transitive Actions on Bruhat-Tits Buildings

There are plenty of discrete transitive actions on an $r$-regular tree, i.e.
rank one Bruhat-Tits tree. However it is rare to have such an action on a
higher
rank Bruhat-Tits building. The objective of this joint, on going work with
“A. Salehi Golsefidy” is to classify all discrete transitive actions on
Bruhat-Tits buildings of higher rank algebraic groups over characteristic
zero
local fields. In this talk we will see how the arithmetic structure of such
lattices and Prasad’s formula for covolume of arithmetic lattices will lead
us
to the following non-existence result; If $F$ is a local field of
characteristic zero then for $n$ larger than 8, there is no discrete
subgroup
of ${\rm PGL}_n(F)$ which acts transitively on the buliding of ${\rm PGL}_n(F)$.
In the positive characteristic case the situation is quite different, in fact
D.~Cartwright and Tim Steger explicitly constructed such actions for any
dimension.