[Paper Review] Maximally symmetric trees

Lee Mosher, Michah Sageev+1 more|ArXiv.org|Dec 1, 2000

Graph theory and applications13 references3 citations

TL;DR

This paper characterizes maximally symmetric trees—bounded valence, bushy, cocompact, thornless trees that realize the 'best' model geometries for virtually free groups. It proves that such trees are uniquely characterized by their isometry groups being maximal uniform cobounded subgroups of their quasi-isometry groups, and provides constructive methods to build them, showing there are countably infinitely many distinct such geometries for the class of virtually free groups of rank ≥2.

ABSTRACT

We characterize the ``best'' model geometries for the class of virtually free groups, and we show that there is a countable infinity of distinct ``best'' model geometries in an appropriate sense--these are the maximally symmetric trees. The first theorem gives several equivalent conditions on a bounded valence, cocompact tree T without valence 1 vertices saying that T is maximally symmetric. The second theorem gives general constructions for maximally symmetric trees, showing for instance that every virtually free group has a maximally symmetric tree for a model geometry.

Motivation & Objective

• To identify the 'best' model geometries for the class of virtually free groups, which are quasi-isometric but lack a common model geometry due to non-isomorphic isometry groups.
• To define and characterize maximally symmetric trees as those whose isometry groups are maximal uniform cobounded subgroups of their quasi-isometry groups.
• To show that such trees are uniquely determined by their isometry group's maximality in embedding into other locally compact groups.
• To provide constructive methods for generating maximally symmetric trees for any virtually free group.
• To demonstrate that there are countably infinitely many distinct isometry types of maximally symmetric trees, resolving the absence of a single universal model geometry for the class.

Proposed method

• Characterize maximally symmetric trees via three equivalent conditions: maximality of Isom(T) in QI(T), uniqueness of embeddings into other isometry groups, and maximality in any locally compact group without compact normal subgroups.
• Use quasi-isometry theory and the structure of bounded valence, bushy, cocompact, thornless trees to analyze isometry group actions and their embeddings.
• Construct maximally symmetric trees via index 1 normalized tree structures and subcover constructions, using edge-indexing and algebraic equations to classify possible isometry types.
• Apply graph-theoretic and group-theoretic techniques to enumerate isometry types of maximally symmetric trees finitely via affine subspaces of integer solutions.
• Use the concept of proper, continuous, cocompact embeddings of Isom(T) into other locally compact groups to define and verify maximality.
• Leverage the fact that any quasi-action on a bounded valence, bushy, thornless tree is quasiconjugate to an isometric action, enabling reduction to isometry group analysis.

Experimental results

Research questions

• RQ1 What conditions characterize a bounded valence, bushy, cocompact, thornless tree as maximally symmetric in terms of its isometry group's maximality in the quasi-isometry group?
• RQ2 Can a unique 'best' model geometry be identified for the class of virtually free groups, given that no single geometry works for all?
• RQ3 How can maximally symmetric trees be constructed explicitly for any virtually free group?
• RQ4 Are there finitely or infinitely many isometry types of maximally symmetric trees, and how can they be enumerated?
• RQ5 What is the role of compact normal subgroups in obstructing maximality of isometry groups, and how are they excluded in the construction?

Key findings

• Maximally symmetric trees are characterized by the property that their isometry groups are maximal uniform cobounded subgroups of their quasi-isometry groups.
• Any continuous, proper, cocompact embedding of the isometry group of a maximally symmetric tree into another locally compact group must be an isomorphism.
• There are countably infinitely many distinct isometry types of maximally symmetric trees, each serving as a unique 'best' model geometry for some virtually free group.
• The construction of maximally symmetric trees is effective and finitistic, using index 1 normalized trees and subcover enumeration via affine subspaces of integer edge-indexing solutions.
• The edge-indexing conditions for maximally symmetric trees are those that avoid lying on any of six specified affine subspaces defined by algebraic equations.
• The paper resolves the absence of a single universal model geometry for the class of virtually free groups by showing that the 'best' geometries are the maximally symmetric trees, of which there are countably many distinct types.