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...



Siegel set in SO(n,1) modulo integer points?
The number of cusps could be more than 1, see [here](http://www.ma.utexas.edu/users/areid/allflat_new.pdf), remark on page 294. 


