  We define matrix groups $FG_n(P)$ for each natural number $n$ and finite set
of primes $P$, such that every rational-valued upper triangular matrix group is
a (possibly distorted) subgroup. Brofferio and Schapira [Brofferio2011poisson],
described the \PF boundary of $GL_n (\mathbb{Q})$ for measures of finite first
moment with respect to adelic length. We show that adelic length is a word
metric estimate on $FG_n(P)$ by constructing another, intermediate, word metric
estimate which can be easily computed from the entries of any matrix in the
group. In particular, finite first moment of a probability measure with respect
to adelic length is an equivalent condition to requiring finite first moment
with respect to word length in $FG_n(P)$.
  We also investigate random walks in the case that $P$ is a length one
sequence. Conditions for pointwise convergence in $\mathbb{R}$ or
$\mathbb{Q}_p$ are given. When these conditions are satisfied, we give path
estimates from boundary points, discuss boundary triviality, show that the
resulting space is a $\mu$-boundary and give cases where the $\mu$-boundary is
the \PF boundary, as conjectured by Kaimanovich.
