Tree lattice subgroups

Abstract

Abstract Let X be a locally finite tree and let G = Aut(X). Then G is naturally a locally compact group. A discrete subgroup Γ ≤ G is called an X-lattice, or a tree lattice if Γ has finite covolume in G. The lattice Γ is encoded in a graph of finite groups of finite volume. We describe several methods for constructing a pair of X-lattices (Γ′, Γ) with Γ ≤ Γ′, starting from ‘edge-indexed graphs’ (A′, i′) and (A, i) which correspond to the edge-indexed quotient graphs of their (common) universal covering tree by Γ′ and Γ respectively. We determine when finite sheeted topological coverings of edge-indexed graphs give rise to a pair of lattice subgroups (Γ, Γ′) with an inclusion Γ ≤ Γ′. We describe when a ‘full graph of subgroups’ and a ‘subgraph of subgroups’ constructed from the graph of groups encoding a lattice Γ′ gives rise to a lattice subgroup Γ and an inclusion Γ ≤ Γ′. We show that a nonuniform X-lattice Γ contains an infinite chain of subgroups Λ1 ≤ Λ2 ≤ Λ3 ≤ ⋯ where each Λ k is a uniform Xk -lattice and Xk is a subtree of X. Our techniques, which are a combination of topological graph theory, covering theory for graphs of groups, and covering theory for edge-indexed graphs, have no analog in classical covering theory. We obtain a local necessary condition for extending coverings of edgeindexed graphs to covering morphisms of graphs of groups with abelian groupings. This gives rise to a combinatorial method for constructing lattice inclusions Γ ≤ Γ′ ≤ H ≤ G with abelian vertex stabilizers inside a closed and hence locally compact subgroup H of G. We give examples of lattice pairs Γ ≤ Γ′ when H is a simple algebraic group of K-rank 1 over a nonarchimedean local field K and a rank 2 locally compact complete Kac–Moody group over a finite field. We also construct an infinite descending chain of lattices ⋯ ≤ Γ2 ≤ Γ1 ≤ Γ ≤ H≤ G with abelian vertex stabilizers.