Maximally symmetric trees

TL;DR

This paper characterizes the most symmetric tree structures for virtually free groups, showing an infinite variety of such maximally symmetric trees and providing conditions and constructions for their existence.

Contribution

It introduces the concept of maximally symmetric trees as model geometries for virtually free groups and provides criteria and methods to construct them.

Findings

There are infinitely many distinct maximally symmetric trees.

Theorems establish equivalent conditions for a tree to be maximally symmetric.

Every virtually free group admits a maximally symmetric tree as a model geometry.

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.