Semidualities from products of trees

TL;DR

This paper introduces the concept of semiduality groups and demonstrates that certain arithmetic groups acting on products of trees are examples, expanding understanding of their algebraic and geometric properties.

Contribution

It defines semiduality groups and shows that groups acting as lattices on products of trees are examples, with additional examples including lamplighter and Diestel-Leader groups.

Findings

$ ext{Gamma}$ is a $ ext{Z}[1/p]$-semiduality group when acting on a product of trees

Examples of semiduality groups include lamplighter, Diestel-Leader, and finite groups

Provides new classes of groups with semiduality properties

Abstract

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