Common overlattices in trees and trees with fins

TL;DR

This paper investigates the extension of a lattice conjugation property from trees to trees with fins, providing a counter-example using generalized universal groups.

Contribution

It demonstrates that the conjugation property for free uniform lattices does not extend to trees with fins, contrasting with previous results for trees.

Findings

Counter-example shows the property fails in trees with fins

Uses generalized universal groups of Burger and Mozes

Highlights limitations of lattice conjugation extension

Abstract

Bass and Kulkarni proved that any pair of free uniform lattices in the automorphism group of a tree have conjugates that both lie inside a third uniform lattice (which is not necessarily free). We show that this does not generalise to trees with fins. The construction of our counter-example involves working with a certain generalisation of the universal groups of Burger and Mozes.