# Fixating Group Actions

## TL;DR

This paper investigates which groups of bijections are fixating, meaning any subgroup with fixed points must have a common fixed point, exploring various geometric and algebraic contexts including Euclidean, elliptic, and hyperbolic spaces.

## Contribution

It characterizes fixating groups of bijections across different geometric spaces and analyzes the influence of space properties and subgroup structures on fixating behavior.

## Key findings

- Euclidean isometry groups are fixating for n ≥ 3
- Certain groups of isometries of elliptic and hyperbolic spaces are fixating
- Existence of free subgroups affects fixating properties

## Abstract

A group of bijections G acting on a set X is said with fixed points (abbreviated as gaf from the french "groupe {\`a} points fixes") if any element of G has at least one fixed point in X. The G group is said with a common fixed point (abbreviated as gag for "groupe {\`a} point fixe global") if there is x $\in$ X fixed by all elements of G. The group G is said fixating if any subgroup of G which is a gaf is automatically a gag. The article explores which groups are fixating. The situation depends on the assumptions made on the group of bijections and on the support set X. For example the group of isometries of the Euclidean space R n is fixating for n 3 but not for n 4. The case of isometries of elliptic and hyperbolic spaces is also considered, as well as that of isometries of some discrete sets. As we will see, the situation depends upon the fact whether the so-called median inequality is satisfied in the ambient space or not. Besides, some of our constructions of nonfixating groups rely on the existence of free subgroups of certain linear groups.