The study of group actions on trees is a cornerstone of geometric group theory. It provides a unified geometric method for working with groups that are free or have an amalgam or HNN structure. Some of the most important examples arise from rank one simple algebraic groups over nonarchimedean local fields acting on their Bruhat-Tits buildings, which are homogeneous or bihomogeneous trees. Other important examples are rank 2 Kac-Moody groups whose Tits buildings are also trees. We may study these groups and their subgroups using the powerful methods of geometric group theory, in particular, the Bass-Serre theory for reconstructing group actions on trees. This gives us a structure theory for constructing these groups in terms of the data from their actions on trees and gives rise to explicit group presentations.
An important class of group actions are discrete groups which, if they act on locally finite trees, act with finite vertex stabilizers. A discrete subgroup of a locally compact group which has finite covolume is called a lattice. Furthermore, a lattice which acts with compact quotient is called a uniform lattice. Lattices appear widely throughout mathematics, particularly the lattice subgroups of Lie groups and their roles in algebraic group theory, geometric group theory and number theory.
Much of our work has been devoted to the study of tree lattices, which are lattice subgroups of automorphism groups of locally finite trees. Lattices in rank one simple algebraic groups over nonarchimedean local fields and rank 2 Kac-Moody groups over finite fields are tree lattices and thus the general Bass-Serre structure theory is applicable in these important cases.
In the early 1980's, Hyman Bass and Alex Lubotzky proposed to study tree lattices in analogy with the remarkable theory of lattices in noncompact semisimple real Lie groups. In joint work with Hyman Bass and Gabriel Rosenberg, we determined conditions that will ensure that automorphism groups of locally finite trees contain tree lattices, giving a positive answer to a conjecture by Bass and Lubotzky.
In another formulation of the question of existence of tree lattices, Bass and Lubotzky conjectured that when uniform lattices are present, under some natural assumptions, there should also be nonuniform lattices in automorphism groups of locally finite trees. This was conjectured in analogy with Borel's theorems in semisimple real Lie groups concerning the coexistence of uniform and nonuniform lattices. We gave a positive answer to this conjecture with a constructive proof, providing a nonuniform tree lattice paired with a uniform one, whenever these natural assumptions are satisfied. In a number of collaborations, we gave necessary and sufficient conditions for automorphism groups of locally finite trees to contain tree lattices, both uniform and nonuniform in all possible settings, answering the Bass-Lubotzky conjectures about existence of tree lattices in full.
We have applied these methods to obtain a number of applications in recent collaborations, for example to studying group actions on higher dimensional hyperbolic buildings, determining fundamental domains for a special class of discrete subgroups known as congruence subgroups of SL(2) in positive characteristic, constructing pairs of tree lattice subgroups and infinite towers of them, connections with finite state automata and decision problems in group theory. We developed a generalization of the Bass-Serre theory for reconstructing group actions on trees to general non-trivial group actions on general sets. Given a knowledge of the point stabilizers of the action and some other natural data, this allows us to give an explicit group presentation for any group acting non-trivially on a set. We have studied the absolute geometry of buildings in the framework of combinatorial arithmetical algebraic geometry. That is, we have obtained a characterization of the core geometry of buildings which is independent of the base field.