  Traditionally, rotation numbers for toroidal billiard flows are defined as
the limiting vectors of average displacements per time on trajectory segments.
The billard trajectories, being curves, oftentimes getting very close to closed
loops, quite naturally define elements of the fundamental group of the billiard
table. The simplest non-trivial fundamental group obtained this way belongs to
the classical Sinai billiard, i.e., the billiard flow on the 2-torus with a
single, convex obstacle removed. This fundamental group is known to be the
group $\textbf{F}_2$ freely generated by two elements, which is a heavily
noncommutative, hyperbolic group in Gromov's sense. We define the homotopical
rotation number and the homotopical rotation set for this model, and provide
lower and upper estimates for the latter one, along with checking the validity
of classically expected properties, like the density (in the homotopical
rotation set) of the homotopical rotation numbers of periodic orbits.
  The natural habitat for these objects is the infinite cone erected upon the
Cantor set $\text{Ends}(\textbf{F}_2)$ of all "ends" of the hyperbolic group
$\textbf{F}_2$. An element of $\text{Ends}(\textbf{F}_2)$ describes the
direction in (the Cayley graph of) the group $\textbf{F}_2$ in which the
considered trajectory escapes to infinity, whereas the height function $t$ ($t
\ge 0$) 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 $\sqrt{2}$, and any direction
$e\in\text{Ends}(F_2)$ for the escape is feasible with any prescribed speed
$s$, $0\leq s\leq \sqrt{2}/2$. This means that the radial upper and lower
bounds for the rotation set $R$ are actually pretty close to each other.
