Trees, unsplittability, property (FA) and the likes

TL;DR

This paper investigates fixed point properties of automorphism groups of countable rooted trees, providing new proofs, conditions for properties like (FA), uncountable strong cofinality, and ample generics, with applications to rigidity results.

Contribution

It offers new proofs and characterizations of fixed point properties in automorphism groups of rooted trees, including conditions for ample generics and implications for rigidity.

Findings

Automorphism groups of rooted trees are unsplittable.

Conditions under which these groups have uncountable strong cofinality.

Necessary and sufficient conditions for having ample generics.

Abstract

The paper is devoted to a study of certain fixed point properties, and their relatives, in the context of full automorphism groups of countable rooted trees. Namely, we study Serre's property (FA'), also called unsplittability, property (FA), the uncountable strong cofinality, and ample generics. We give a new proof of a theorem of Psaltis to the extent that automorphism groups of rooted trees are unsplittable, and show under what circumstances automorphism groups of rooted trees have uncountable strong cofinality (and thus property (FA).) Also, we prove a necessary and sufficient condition for automorphism groups of rooted trees to have ample generics. This very strong property has interesting implications, such as the small index property, and continuity of homomorphisms. As an application, we analyze the relationship between two generalizations (discovered by Psaltis and Forester) of an interesting rigidity result on locally finite trees, originally proved by Bass and Lubotzky.