Traditionally, rotation numbers for toroidal billiard flows are defined as
the limiting vectors of average displacements per time on trajectory segments.
Naturally, these creatures are living in the (commutative) vector space
$\real^n$, if the toroidal billiard is given on the flat $n$-torus.
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 classicaly 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.