Semidualities from products of trees

Abstract

Let $K$ be a global function field of characteristic $p$, and let $\mathrm{\Gamma }$ be a finite-index subgroup of an arithmetic group defined with respect to $K$ and such that any torsion element of $\mathrm{\Gamma }$ is a $p$–torsion element. We define semiduality groups, and we show that $\mathrm{\Gamma }$ is a $\mathbb{Z}\left[1∕p\right]$–semiduality group if $\mathrm{\Gamma }$ 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.