Introduction

Abstract

Let X be a locally finite tree. Then G = Aut(X) is a locally compact group. The vertex stabilizers Gx are open and compact (in fact, profinite). A subgroup Γ ≤ G is discrete if Γx is finite for some, and hence every vertex x∈VX. In this case we define $$Vol(\Gamma \backslash \backslash X): = \sum\limits_{x\varepsilon \Gamma \backslash VX} {\frac{1}{{|\Gamma _x |}},}$$ (1)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.