Title: Maximal entries of elements in certain matrix monoids,
Abstract: Let $L_u=\begin{bmatrix}1 & 0\\u & 1\end{bmatrix}$ and $R_v=\begin{bmatrix}1
& v\\0 & 1\end{bmatrix}$ be matrices in $SL_2(\mathbb Z)$ with $u, v\geq 1$.
Since the monoid generated by $L_u$ and $R_v$ 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 $n\geq 1$. By using ideas from our
previous work on $(u,v)$-Calkin-Wilf trees, we extend their results for any $u,
v\geq 1$ 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 $L_u$ and $R_v$.