Lattices in automorphism groups of trees
Degree GrantorUniversity of Canterbury
Lattices are intrinsically interesting mathematical objects that arise in diverse and seemingly disparate subfields of pure mathematics. Due to the work of Bass, Serre, and others, it has become clear that one way of investigating these objects is to analyse hom(Λ, G) and hom(Λ, G)/G, respectively, the set of all monomorphisms from Λ to G, and the set of orbits under the action of G where A is a fundamental group of a graph of groups and G = GL₂(C). We shall concern ourselves with the case where A is derived from a graph of groups consisting of a single vertex and a single loop. In particular, when considering the quotient space, hom(Λ, G)/G , the questions concerning us are: What is the dimension of hom(Λ, G)/G? Is hom(Λ, G)/G smooth? These questions will be answered for two particular Λ that arise in the aforementioned way. Along the way we provide the necessary background theory in group presentations, graphs of groups, lattices, dimension, and smoothness. In what follows we draw on group theory and linear algebra frequently. With respect to the latter, we assume the reader is familiar with the diagonalization process and the closely related topics of eigenvalues and eigenvectors. If not, the relevant material can be found in [5, ch. 4.4]. Acquaintance with group actions, orbits and stabilizers, free groups, and the first isomorphism theorem is assumed regarding group theory. These topics can be found in .
SubjectsField of Research::01 - Mathematical Sciences
- Engineering: Reports