0. Introduction

Let G be a locally compact group, and \i a left invariant Haar measure on G. A

discrete subgroup V of G is called a G- lattice if fj,(T\G) is finite, and a uniform (or

cocompact) G-lattice if T\G is compact, non-uniform otherwise.

(1) In the early 1980's, Hyman Bass and Alex Lubotzky proposed to study lattices in

the automorphism group of a locally finite tree X, a group that is naturally locally com-

pact, in analogy with lattices in non-compact simple real Lie groups. While G — Aut(X)

is not simple, Tits has shown ([Ti]) that when G acts minimally on X, fixing no end of

X, then G has a large simple normal subgroup, G + , generated by all edge stabilizers. In

fact, when X is homogeneous, G+ is of index two, so G is 'almost simple'.

(2) The program of Bass and Lubotzky was motivated by the intermediate case of a

simple algebraic K-group H, of K-r&nk 1, over a non-archimedan local field K, with finite

residue field ¥q. The group H Aut(X) acts on its Bruhat-Tits tree X\ for example, if

H — PSL2{K) then X is the homogeneous tree Xq+\.

(3) With respect to this program of study, many of the natural questions have been

treated. Some examples are: the existence of uniform tree lattices ([BK]), and of non-

uniform tree lattices (the present work, and [BCR], [C2], [CR1]), the structure of uniform

and non-uniform tree lattices ([BK], [BL]), covolumes ([BK], [BL], [IL], [R]), commen-

surability groups of uniform tree lattices ([BK], [YL]), super-rigidity ([LMZ], [BM]), the

congruence subgroup problem for uniform lattices on regular trees ([Mo]), and the exis-

tence of towers of lattices ([CR2], [CR3], [R]).

(4) R. Kulkarni, in [K], has also indicated an analogy of the study of trees and their

lattices with the study of discontinuous groups and Riemann surfaces, as well as direct

connections with automorphisms of graphs, free groups and surfaces, and with the struc-

ture of finite groups.

(5) For the study of tree lattices in analogy with lattices in Lie groups, many interesting

questions as yet remain open, such as the arithmeticity and commensurability of non-

uniform tree lattices, the congruence subgroup problem on non-homogeneous trees, and

Received by the editor January 27, 1998

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