Fellow traveller quasi-axes

Description

This article describes a equivalence relation between Quasi-axis which I call fellow traveller for lack of a better term. Two quasi-axes are called fellow traveller if there are group elements which move the two axes together for any length of time. The name fellow traveller comes from Fellow travellers in combinatorial group theory where two groups are called fellow travellers, if their paths in the cayley graph do not diverge to far from another.

• Is fellow traveller a good term?

Transfer orientation to other geodesic arcs

Let $J$ be a sufficiently long $(k,ϵ)$-Quasi-geodesic segment inside the $K(k,ϵ,\delta )$-neighbourhood (of the Morse lemma) of a $(k,ϵ)$-Quasi-axis $l$. We call points in $l$ ahead of other points in $l$ when they lie downstream (in relation to the orientation). For the start and terminal point of $J$ we can choose two points in $l$. If the point near the terminal point is ahead of the other, this induces a $g$-orientation on $J$. Of course, this only works, if $J$ is long enough. For short segments, it might not be clear how to determine whether start of terminal is more ahead

Definition Fellow traveller axes

When $g_1$ and $g_2$ are hyperbolic elements of $G$ we will write $g_1\sim g_2$ if for an arbitrarily long segment $J$ in a $(k,ϵ)$-quasi-axis for $g_1$ there is $g\in G$ such that $g(J)$ is within the $B(k,ϵ,\delta )$ neighbourhood of a $(k,ϵ)$-quasi-axis of $g_2$. Furthermore the $g_2$-orientation should be equal to the induced orientation of $g(J)$ given by $g_1$.

This is not the same as parallelity as in euclidean space. Here, there doesn’t need to be an element that make the two axes go parallel forever. Instead, we can make the two axes walk longer and longer together by choosing a suitable sequence of elements $g_i$.

Properties

We notice, that in our definition of parallel axes we used the neighbourhood $B(k,ϵ,\delta )$. Choosing any bigger neighbourhood-radius wouldn’t change much. if two quasi-geodesics are $C$-close for any length then that must mean that they are only a finite distance apart and due to the Morse lemma they are already $B(k,ϵ,\delta )$ apart. I think.

But I don’t understand yet, why we can drop the $(k,ϵ)$. lit_bestvinaBoundedCohomologySubgroups2002, p.72

• Why can we drop the $(k,ϵ)$ here?

Characterisation parallel axes

We have the following information on the parallel axes:

• $\sim$ is an equivalence relation
• $g_1\sim g_2$ if and only if $g_1^k\sim g_2^l$ for any $kl>0$
• If $g_1$ and $g_2$ have positive powers which are conjugate in $G$ then $g_1\sim g_2$

If Weak proper discontinuity holds, then the converse of the third bullet holds as well.

Examples

Example