2018/03/27 by Angel Pardo, Pardo, Angel
Mathematics · #FOS: Mathematics #Group Theory (math.GR) #Spectral Theory (math.SP) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1804.00537
openalex publication_date 2018/03/27 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
We compute the cone types of the Cayley graph of the modular group PSL(2,Z) associated with the standard system of generators \small\(\beginsmallmatrix 0 & -1 1 & 0 \endsmallmatrix),(\beginsmallmatrix 1 & 1 0 & 1 \endsmallmatrix)\. 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.