# Comparison of volumes of Siegel sets and fundamental domains for \(\mathrm {SL}_n (\pmb {\mathbb {Z}})\)

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## Abstract

The purpose of this paper is to calculate explicitly the volumes of Siegel sets which are coarse fundamental domains for the action of \({\mathrm {SL}} _n (\mathbb {Z})\) in \(\mathrm {SL} _n (\mathbb {R})\), so that we can compare these volumes with those of the fundamental domains of \({\mathrm {SL}} _n (\mathbb {Z})\) in \(\mathrm {SL} _n (\mathbb {R})\), which are also computed here, for any \(n\ge 2\). An important feature of this computation is that it requires keeping track of normalization constants of the Haar measures. We conclude that the ratio between volumes of fundamental domains and volumes of Siegel sets grows super-exponentially fast as *n* goes to infinity. As a corollary, we obtained that this ratio gives a super-exponencial lower bound, depending only on *n*, for the number of intersecting Siegel sets. We were also able to give an upper bound for this number, by applying some results on the heights of intersecting elements in \( {\mathrm {SL}} _n (\mathbb {Z})\).

## Keywords

Arithmetic groups Siegel sets Coarse fundamental domains Volumes## Mathematics Subject Classification

20G20 20G30 51N30 14L35## Notes

### Acknowledgements

I would like to thank Professor Mikhail Belolipetsky for several suggestions on the development of this paper and also on the text. I also thank Paul Garret and Martin Orr for their very helpful works and for always answering my emails with good suggestions, and Cayo Dória for helping me to understand better some topics. Finally, I also thank the refferee for carefully reading the paper and for giving suggestions that improved the presentation of the results.

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