Common overlattices in trees and trees with fins

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.

arXiv comment18 pages, 3 figures; v2: Some minor changes according to the referee's comments. To appear in the Journal of Algebra