Cone types and asymptotic invariants for the random walk on the modular group

Abstract

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. Gouëzel (2015). Finally, using results of Y. Guivarc'h (1980) and S. Gouëzel, F. Mathéus and F. Maucourant (2015), we estimate other asymptotic invariants of the random walk, namely, the entropy and the drift.