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Tree lattice subgroups

Tree lattice subgroups 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 Λ 1 ≤ Λ 2 ≤ Λ 3 ≤ ⋯ where each Λ k is a uniform X k -lattice and X k 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 ⋯ ≤ Γ 2 ≤ Γ 1 ≤ Γ ≤ H ≤ G with abelian vertex stabilizers. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Groups - Complexity - Cryptology de Gruyter

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Publisher
de Gruyter
Copyright
Copyright © 2011 by the
ISSN
1867-1144
eISSN
1869-6104
DOI
10.1515/gcc.2011.001
Publisher site
See Article on Publisher Site

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 Λ 1 ≤ Λ 2 ≤ Λ 3 ≤ ⋯ where each Λ k is a uniform X k -lattice and X k 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 ⋯ ≤ Γ 2 ≤ Γ 1 ≤ Γ ≤ H ≤ G with abelian vertex stabilizers.

Journal

Groups - Complexity - Cryptologyde Gruyter

Published: May 1, 2011

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