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Tree lattice subgroups

Tree lattice subgroups Let X be a locally finite tree and let G == Aut( X ). Then G is naturally a locally compact group. A discrete subgroup ΓΓ ≤≤ G is called an X-lattice , or a tree lattice if ΓΓ has finite covolume in G . The lattice ΓΓ is encoded in a graph of finite groups of finite volume. We describe several methods for constructing a pair of X -lattices (ΓΓ′′, ΓΓ) with ΓΓ ≤≤ ΓΓ′′, 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 ΓΓ′′ and ΓΓ respectively. We determine when finite sheeted topological coverings of edge-indexed graphs give rise to a pair of lattice subgroups (ΓΓ, ΓΓ′′) with an inclusion ΓΓ ≤≤ ΓΓ′′. We describe when a ‘‘full graph of subgroups’’ and a ‘‘subgraph of subgroups’’ constructed from the graph of groups encoding a lattice ΓΓ′′ gives rise to a lattice subgroup ΓΓ and an inclusion ΓΓ ≤≤ ΓΓ′′. We show that a nonuniform X -lattice ΓΓ contains an infinite chain of subgroups णΛ 1 ≤≤ णΛ 2 ≤≤ णΛ 3 ≤≤ ⋯⋯ where each णΛ 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 ΓΓ ≤≤ ΓΓ′′ ≤≤ H ≤≤ G with abelian vertex stabilizers inside a closed and hence locally compact subgroup H of G . We give examples of lattice pairs ΓΓ ≤≤ ΓΓ′′ 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 ⋯⋯ ≤≤ ΓΓ 2 ≤≤ ΓΓ 1 ≤≤ ΓΓ ≤≤ 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
©© de Gruyter 2011
ISSN
1867-1144
eISSN
1869-6104
DOI
10.1515/GCC.2011.001
Publisher site
See Article on Publisher Site

Abstract

Let X be a locally finite tree and let G == Aut( X ). Then G is naturally a locally compact group. A discrete subgroup ΓΓ ≤≤ G is called an X-lattice , or a tree lattice if ΓΓ has finite covolume in G . The lattice ΓΓ is encoded in a graph of finite groups of finite volume. We describe several methods for constructing a pair of X -lattices (ΓΓ′′, ΓΓ) with ΓΓ ≤≤ ΓΓ′′, 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 ΓΓ′′ and ΓΓ respectively. We determine when finite sheeted topological coverings of edge-indexed graphs give rise to a pair of lattice subgroups (ΓΓ, ΓΓ′′) with an inclusion ΓΓ ≤≤ ΓΓ′′. We describe when a ‘‘full graph of subgroups’’ and a ‘‘subgraph of subgroups’’ constructed from the graph of groups encoding a lattice ΓΓ′′ gives rise to a lattice subgroup ΓΓ and an inclusion ΓΓ ≤≤ ΓΓ′′. We show that a nonuniform X -lattice ΓΓ contains an infinite chain of subgroups णΛ 1 ≤≤ णΛ 2 ≤≤ णΛ 3 ≤≤ ⋯⋯ where each णΛ 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 ΓΓ ≤≤ ΓΓ′′ ≤≤ H ≤≤ G with abelian vertex stabilizers inside a closed and hence locally compact subgroup H of G . We give examples of lattice pairs ΓΓ ≤≤ ΓΓ′′ 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 ⋯⋯ ≤≤ ΓΓ 2 ≤≤ ΓΓ 1 ≤≤ ΓΓ ≤≤ H ≤≤ G with abelian vertex stabilizers.

Journal

Groups - Complexity - Cryptologyde Gruyter

Published: May 1, 2011

Keywords: Tree lattice; lattice subgroups; automorphism groups of trees

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