Non-nesting actions of Polish groups on real trees

TL;DR

This paper proves that Polish groups with a comeagre conjugacy class acting non-nestingly on real trees must have all elements fixing a point, revealing a strong fixed point property under these conditions.

Contribution

It establishes a new fixed point property for a class of Polish groups acting on real trees, linking group dynamics with topological properties.

Findings

Groups with comeagre conjugacy classes have fixed points in non-nesting actions.

Non-nesting actions on R-trees imply point fixing for all group elements.

Provides conditions under which Polish groups exhibit fixed point properties.

Abstract

We prove that if a Polish G with a comeagre conjugacy class has a non-nesting action on an R-tree, then every element of G fixes a point.