  We compute the cone types of the Cayley graph of the modular group
$\mathrm{PSL}(2,\mathbf{Z})$ associated with the standard system of generators
${\small\left\{\left(\begin{smallmatrix} 0 & -1 \\ 1 & 0
\end{smallmatrix}\right),\left(\begin{smallmatrix} 1 & 1 \\ 0 & 1
\end{smallmatrix}\right)\right\}}$. We do this by showing that, in general,
there is a set of suffixes of each element that completely determines the cone
type of the element, and such suffixes are subwords of primitive relators.
  Then, using J. W. Cannon's seminal ideas (1984), we compute its growth
function. We estimate from above and below the spectral radius of the random
walk using ideas from T. Nagnibeda (1999) and S. Gou\"ezel (2015). Finally,
using results of Y. Guivarc'h (1980) and S. Gou\"ezel, F. Math\'{e}us and F.
Maucourant (2015), we estimate other asymptotic invariants of the random walk,
namely, the entropy and the drift.
