I wonder what is known about a fundamental region for SO($n,1$) modulo its integer points? is there only one cusp? and if one writes a Siegel set in the form of
$K A_\tau N_c$, where $N_c$ is compact and $A_\tau$ is the set of elements diag$(a,1,...,1,1/a)$ for $a < \tau$, then what is an estimate for $\tau$? in SL$_n$ one has $2/\sqrt{3}$, will it work for SO$(n,1)$ for any $n$?
Any reference will be appreciated. Maybe even a precise fundamental domain in $H^n$ is known for this example? then it should not be difficult to see how big a Siegel set needs to be to contain this fundamental domain...