  In this paper we study the homotopical rotation vectors and the homotopical
rotation sets for the billiard flow on the unit flat torus with three, mutually
intersecting and mutually orthogonal cylindrical scatterers removed from it.
  The natural habitat for these objects is the infinite cone erected upon the
Cantor set $\text{Ends}(\textbf{F}_3)$ of all "ends" of the hyperbolic group
$\textbf{F}_3=\pi_1(\mathbf{Q})$. An element of $\text{Ends}(\textbf{F}_3)$
describes the direction in (the Cayley graph of) the group $\textbf{F}_3$ in
which the considered trajectory escapes to infinity, whereas the height
function $s$ ($s\ge 0$) of the cone gives us the average speed at which this
escape takes place.
  The main results of this paper claim that the orbits can only escape to
infinity at a speed not exceeding $\sqrt{3}$, and in any direction
$e\in\text{Ends}(\mathbf{F}_3)$ the escape is feasible with any prescribed
speed $s$, $0\leq s\leq 1/3$. This means that the radial upper and lower bounds
for the rotation set $R$ are actually pretty close to each other. Furthermore,
we prove the convexity of the set $AR$ of constructible rotation vectors, and
that the set of rotation vectors of periodic orbits is dense in $AR$. We also
provide effective lower and upper bounds for the topological entropy of the
studied billiard flow.
