Nonelementary action

Description

An action of a group on a Hyperbolic metric space is called nonelementary if there are at least two Hyperbolic group elements whose Quasi-axis do not contain rays which are a finite distance from one another.

Nonelementary groups are characterised by two directions in which one can move.

Definition

An action of a group $G$ on a Hyperbolic metric space $X$ is called nonelementary if there are at least two Hyperbolic group elements $g_1,g_2$whose Quasi-axis $l_1,l_2$ do not contain rays which are a finite distance from one another. These two elements are called independent.

Properties

Infinitedimensional set of Quasi-morphisms

Let $G$ be a group acting on a $\delta$-hyperbolic graph $X$ by isometries. If the group is nonelementary and there exist independent elements $g_1,g_2$ which are not parallel, meaning $g_1\nsim g_2$. Then the Coarse quasi-homomorphism space $\widetilde{QM(G)}$ is infinite-dimensional.

*How do independent elements which are parallel look like? Well, you might have two axis which can be put next to another but not for eternity. Instead you use a sequence of group elements to make them travel together more and more. There is also the opposite case. But this is easily answered by $f\nsim f^{-1}$ which has the same quasi-axis but they do not have the same orientation.

Existence of infinitely many-nonparallel elements

Let $G$ be a group acting on a $\delta$-hyperbolic graph $X$ by isometries. If the group is nonelementary and there exist independent elements $g_1,g_2$ which are not parallel, meaning $g_1\nsim g_2$. Then there is an infinite sequence $f_1,f_2,...\in G$ of Hyperbolic group elements such that

• $f_i\nsim f_i^{-1}$ for all $i$
• $f_i\nsim f_j^{\pm 1}$ for all $j<i$

Examples

Special cases of the infinitedimensional Quasi-morphism theorem