Semidualities from products of trees

Let K be a global function field of characteristic p, and let Γ be a finite-index subgroup of an arithmetic group defined with respect to K and such that any torsion element of Γ is a p-torsion element. We define semiduality groups, and we show that Γ is a ℤ [1∕p]-semiduality group if Γ acts as a lattice on a product of trees. We also give other examples of semiduality groups, including lamplighter groups, Diestel-Leader groups, and countable sums of finite groups.