Maximal entries of elements in certain matrix monoids

Abstract

Let Lu = 1 0 u 1 and Rv = 1 v 0 1 be matrices in SL2(Z) with u, v ≥ 1. Since the monoid generated by Lu and Rv is free, we can associate a depth to each element based on its product representation. In the cases where u = v = 2 and u = v = 3, Bromberg, Shpilrain, and Vdovina determined the depth n matrices containing the maximal entry for each n ≥ 1. By using ideas from our previous work on (u, v)-Calkin-Wilf trees, we extend their results for any u, v ≥ 1 and in the process we recover the Fibonacci and some Lucas sequences. As a consequence we obtain bounds which guarantee collision resistance on a family of hashing functions based on Lu and Rv.