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

TL;DR

This paper analyzes the structure of the Cayley graph of the modular group, computes its growth function, and estimates key asymptotic invariants of the associated random walk, including spectral radius, entropy, and drift.

Contribution

It introduces a method to determine cone types via suffixes of elements and applies classical and recent theories to compute growth and asymptotic invariants for the modular group.

Findings

Computed cone types using suffixes of elements.

Estimated spectral radius bounds for the random walk.

Calculated asymptotic invariants like entropy and drift.

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. 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.