Structural properties of dendrite groups

TL;DR

This paper investigates the structure of the homeomorphism group of dendrites, revealing the existence of numerous simple groups with fixed point properties and analyzing their normal subgroups.

Contribution

It identifies uncountably many non-isomorphic simple groups within the homeomorphism group of dendrites and studies their action properties.

Findings

Existence of uncountably many non-isomorphic simple groups G

These groups have the fixed point property (FH) for isometric actions

Normal subgroup structure of the homeomorphism group of dendrites

Abstract

Let G be the homeomorphism group of a dendrite. We study the normal subgroups of G. For instance, there are uncountably many non-isomorphic such groups G that are simple groups. Moreover, these groups can be chosen so that any isometric G-action on any metric space has a bounded orbit. In particular they have the fixed point property (FH).