Tree lattice subgroups

Abstract

is finite, and a uniform X-lattice if Γ\X is a finite graph, non-uniform otherwise. In [C2] and [BCR] we gave the necessary and sufficient conditions for the existence of X-lattices. In this work we determine how to construct pairs of tree lattice subgroups. In the setting of topological covering theory, there is a correspondence between coverings, p : X → A, and subgroups of π1(A), and this gives essentially ‘one’ subgroup up to conjugacy. Here with lattices given by their quotient graphs of groups, we are in an ‘orbifold’ setting where coverings have an extra ingredient, namely isotropy groups. We make use of this additional data to construct lattice subgroups by using subgroups of isotropy (vertex) groups. This will give rise to zero, one, finitely many, or infinitely many possible subgroups up to isomorphism. In fact in [CR1] the authors exhibited infinite ascending chain of lattice subgroups, all with the same quotient graph. We describe several methods (sections 3-5) for constructing a pair of Xlattices (Γ′,Γ) with Γ ≤ Γ′, starting from ‘edge-indexed graphs’ (A′, i′) and (A, i) which will correspond to the edge-indexed quotient graphs of their (common) universal covering tree by Γ′ and Γ respectively. Our techniques are a combination of topological graph theory, covering theory for graphs of groups ([B]), and covering theory for edge-indexed graphs developed in [C1] and [BCR]. As an application, we show (section 5) that a nonuniform X-lattice Γ contains an infinite chain of subgroups Λ1 < Λ2 < Λ3 < . . . where each Λk is a uniform Xj-lattice, Xk a subtree of X. Let φ : (A, i) → (A′, i′) be a covering of edge-indexed graphs (defined in section 2). We wish to determine if it is possible to extend φ to a covering