I am reading "The Tits alternative for $\operatorname{Out}(F_n)$ I: Dynamics of exponentially-growing automorphisms" by Mladen Bestvina et al. for my Master's thesis. On page 526, the following part can be found: 
I have some questions concerning this part of the text:
- "The circle at infinity is replaced by the Cantor set $\partial F_n$ of ends of $F_n$. I looked up what the "end" of a group is and got redirected to the wikipedia page about the "end of a graph". What I understood from this is that an end of a graph is an equivalence class of lines and that the ends of a group are the ends of the associated Cayley graphs. However, I do not understand there is a unique line $\tilde{\sigma} \subset \Gamma$ connecting the ends $c_1$ and $c_2$
- What is the the diagonal action of $F_n$ on $\partial F_n \times \partial F_n$? I know what a diagonal action of a group on a set is, but what is the action of $F_n$ on its boundary?
- What are covering translations? I know what covering transformations are, so is this just the same thing denoted by a different name?
Suppose $X$ is a (locally path connected) topological space that has an exhaustion $K_1\subset K_2\subset\dots \subset X$ by compact sets. Each complement $X-K_i$ has some number of connected components, written $\pi_0(X-K_i)$. When $i\geq j$, there is an induced map $f_{ij}:\pi_0(X-K_i)\to \pi_0(X-K_j)$ that "fuses" connected components. An end of $X$ is an element of the inverse limit of this system. More concretely, an end is a sequence $\{U_i\}_i$ with $U_i\in\pi_0(X-K_i)$ and $f_{ij}(U_i)=U_j$. More intuitively, an end is a sequence of more-and-more-refined connected components in the complement of a closed ball of ever-increasing radius. We topologize the space of ends by declaring, for each open $V\subset X$, that the set of all ends that eventually lie in $V$ is open
For a finitely generated group $G$, we define the ends of $G$ to be the ends of its Cayley graph (or more generally the ends of a topological space that $G$ acts "geometrically" on, if I recall correctly what I've heard at various geometric group theory talks I've attended; this means "properly" and "cocompactly"). It turns out it doesn't matter which generating set you choose: there will be a homeomorphism between the ends of the first Cayley graph to the ends of the second Cayley graph.
The ends of $F_n$ are straightforward to describe. If we choose the natural generating set of $F_n$, the Cayley graph $\Gamma$ is a tree where every vertex is degree $2n$. Let $K_i$ be the set of all points that are within distance $i$ of the identity's vertex, where the distance is given by saying each edge has unit length. Connected components of $\Gamma-K_i$ can be described as paths from the identity's vertex, and in particular there is a unique reduced word in $F_n$ of minimal length that corresponds to such a path. The ends, then, are described by "infinitely long reduced words" in $F_n$, for instance $abababa\dotsc,$ which should be thought of as a limit of finite reduced words. The topology on the ends is the same as having a basis indexed by reduced words $w$ in $F_n$ of all infinite reduced words that have $w$ as a prefix. This is like the product topology on $\{0,1\}^{\mathbb{N}}$, and in both cases they are homeomorphic to the Cantor set.
These infinite reduced words are the "lines" or "geodesics" in the Cayley graph of $F_n$. Because the Cayley graph is a connected tree, there is a unique non-backtracking line between any two points. A line between ends can be described entirely by the two end-points. This is $\partial F_n\times \partial F_n$. But the end-points must be distinct, hence we subtract the diagonal. But also we don't care about the direction of the line, so we mod by the $Z_2$ action.
Covering translations are the same as covering transformations. I think the point is that every line between ends can be translated by covering transformations, but what happens to the end-points is described by the $F_n$ action on $\partial F_n$. Furthermore, every line can be described by a pair of reduced infinite words that each start from the identity's vertex, with the understanding that they might have a common prefix, representing the exact covering transformation whose inverse would bring that line to the identity's vertex.