  Linear and projective boundaries of Cayley graphs were introduced
in~\cite{kst} as quasi-isometry invariant boundaries of finitely generated
groups. They consist of forward orbits $g^\infty=\{g^i: i\in \mathbb N\}$, or
orbits $g^{\pm\infty}=\{g^i:i\in\mathbb Z\}$, respectively, of non-torsion
elements~$g$ of the group $G$, where `sufficiently close' (forward) orbits
become identified, together with a metric bounded by 1.
  We show that for all finitely generated groups, the distance between the
antipodal points $g^\infty$ and $g^{-\infty}$ in the linear boundary is bounded
from below by $\sqrt{1/2}$, and we give an example of a group which has two
antipodal elements of distance at most $\sqrt{12/17}<1$. Our example is a
derivation of the Baumslag-Gersten group. \newline We also exhibit a group with
elements $g$ and $h$ such that $g^\infty = h^\infty$, but $g^{-\infty}\neq
h^{-\infty}$. Furthermore, we introduce a notion of
average-case-distortion---called growth---and compute explicit positive lower
bounds for distances between points $g^\infty$ and $h^\infty$ which are limits
of group elements $g$ and $h$ with different growth.
