Tree Lattices pp 1-12 | Cite as

# Introduction

- 354 Downloads

## Abstract

*X*be a locally finite tree. Then

*G*= Aut(

*X*) is a locally compact group. The vertex stabilizers

*G*

_{x}are open and compact (in fact, profinite). A subgroup Γ ≤

*G*is

*discrete*if Γ

_{x}is finite for some, and hence every vertex

*x∈VX*. In this case we define

call Γ an *X-lattice* if Vol(Γ\\*X*) < ∞, and call Γ a uniform *X-lattice* if Γ\*X* is a finite graph. In case *G*\*X* is finite, this is equivalent to Γ being a lattice (resp., uniform lattice) in the locally compact group *G*. These tree lattices are the object of study in this work. The technique used is the theory of graphs of groups ([S], Ch. I), as elaborated in [B3]. The study of uniform tree lattices was initiated in [BK]. The present work, in some ways a sequel to [BK], focuses much more on the non-uniform case. Here the phenomena are much more complex and varied. Accordingly, we devote a great deal of attention to the construction and analysis of diverse examples.

## Keywords

Compact Group Tree Lattice Finite Graph Uniform Tree Uniform Lattice## Preview

Unable to display preview. Download preview PDF.