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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...

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I would suggest looking at Chapter 28 of sphere packings lattices and groups: books.google.com/… –  Ian Agol Mar 8 '13 at 20:56
    
The number of cusps is the number of Euclidean unimodular lattices of dimension $n-1$, which grows quite rapidly: en.wikipedia.org/wiki/Unimodular_lattice –  Ian Agol Mar 8 '13 at 22:02
    
Thank you very much! –  D.Kleinbock Mar 8 '13 at 22:20
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up vote 2 down vote accepted

The number of cusps could be more than 1, see here, remark on page 294.

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Thanks! I was not aware of that. And yet I wonder, when there is only one cusp, can one say anything about a bound for $\tau$? –  D.Kleinbock Mar 8 '13 at 21:59
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