Definition of hyperbolic elements and axes in a (limit) group

Question

I am writing on my master thesis at the moment and it is based on the following article of Fujiwara and Sela: https://arxiv.org/abs/2002.10278

On page 7 they use the terms "hyperbolic element" and their "axes". I searched for a definition on the internet but I only found one for the case of SL2(R).

What is the general definition of these two terms in a group (or if such a general definition does not exist, what is the definition for a limit group etc.)? It would also be nice if you could give me a reference for these definitions because I need a source to cite for my master thesis.

Answer 1

When a group $G$ acts by isometries on a hyperbolic space $X$, there are three kinds of elements, classified by their fixed point structure on the union of $X$ with its boundary $\partial X$. An element is:

1. elliptic if it fixes a point of $X$, or
2. hyperbolic if it fixes no point of $X$ and exactly two points of $\partial X$, or
3. parabolic if it fixes no point of $X$ and a single point of $\partial X$.

In this setting, $\operatorname{PSL}_2(\mathbb{R})$ is particularly important as it is the group of orientation-preserving isometries of the hyperbolic plane $\mathbb{H}^2$; this is likely why you found the internet searches you did.

For a hyperbolic element, the two points on the boundary are connected by an "axis", which the element acts along by translation.

In geometric group theory we often replace hyperbolic spaces with metric spaces which have hyperbolic-like properties, such as $\delta$-hyperbolic or $\operatorname{CAT}(0)$ spaces. From this point of view, the following definition is often used.

Let $G$ be a group acting by isometries on a metric space $X$. The displacement function of $g\in G$ is the function $X\to\mathbb{R}^*$ defined by $d_g(x):= d(gx, x)$. The translation length of $g$ is the number $|g|:=\inf\{d_g(x)\mid x\in X\}$. Then an element $g\in G$ is:

1. elliptic if it fixes a point of $X$,
2. hyperbolic if $d_g$ attains a strictly positive minimum (i.e. there exists $x \in X$ such that $|g|=d_g(x)$ and $|g|>0$), or
3. parabolic if its minimum is never attained.

Here, the axis of a hyperbolic element is the set of points which attain the minimum, so the set $\operatorname{Min}(g)=\{x\in X\mid d_g(x)=|g|\}$.

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In the linked article, the groups are limit groups of a fixed hyperbolic group, and the paper analyzes the action of these limit groups on limit trees. These limit trees are hyperbolic-like, so we may apply the above ideas.

For groups acting on trees, elements are never parabolic (as all distances are natural numbers). However, from my understanding, limit trees are not actually trees, but $\mathbb{R}$-trees; see wikipedia or these notes of Wilton. This means that parabolic elements may exist.

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For a reference, Bridson and Haefliger's book Metric spaces of non-positive curvature (link) contains much of this, in particular their Chapter 11.6.