Higher-dimensional Dehn functions for S-arithmetic groups

A group $\Gamma$ if of type $F_k$ if it admits an
Eilenberg MacLane complex with finite k-skeleton.
For such groups one can define the (k-1)-dimensional Dehn function,
which measures the difficulty to fill (k-1)-cycles by k-chains.
 I will describe the optimal higher-dimensional Dehn functions for uniform
 S-arithmetic subgroups of reductive groups over global fields.
 I will also discuss a conjectural picture for non-uniform S-arithmetic
 groups.