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I'm looking for a reference and/or the original source for the following fact:

An irreducible non-uniform lattice in a semisimple Lie group without compact factor has Q-rank 1 if and only if it does not contain a subgroup isomorphic to a finite index subgroup of $SL(3,{\Bbb Z})$ or $SO(2,3)_{\Bbb Z}$.

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  • $\begingroup$ The question should boil down to classifying (up to isogeny) minimal $\mathbf{Q}$-simple groups of $\mathbf{Q}$-rank $\ge 2$ and show that these are the $\mathbf{Q}$-split forms of $SO_5$ and $SL_3$. $\endgroup$
    – YCor
    Oct 7, 2013 at 13:35
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    $\begingroup$ Concerning the easy implication, probably superrigidity is enough, but I guess it can be avoided: for instance showing that $\mathbf{Q}$-rank $\le 1$ implies that every polycyclic subgroup is virtually nilpotent, while both of $SL_3(\mathbf{Z})$ and $SO(3,2)_\mathbf{Z}$ have polycyclic subgroups of exponential growth. $\endgroup$
    – YCor
    Oct 7, 2013 at 13:51
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    $\begingroup$ This appears in Witte-Morris' book about arithmetic-groups - people.uleth.ca/~dave.morris/books/IntroArithGroups.pdf, see ch9 section H in there. $\endgroup$
    – Asaf
    Oct 7, 2013 at 15:06
  • $\begingroup$ Unfortunately that chapter 9 is unfinished and has no references yet. $\endgroup$
    – ThiKu
    Oct 8, 2013 at 5:18
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    $\begingroup$ I think it is in Margulis' book. $\endgroup$
    – Misha
    Oct 8, 2013 at 12:05

1 Answer 1

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The proof of Kazhdan's property (T) for real simple Lie groups of real rank at least two as given in the old Bourbaki talk of Kirilllov and Delaroche involves showing (property (T) for $H=SL_3({\mathbb R}), Sp_2({\mathbb R})$ and then showing) that any such $G$ contains a subgroup locally isomorphic to $H$.

Exactly the same proof shows that any $\mathbb Q$-simple linear algebraic group $G$ of $\mathbb Q$ rank at least two contains a subgroup locally $\mathbb Q$-isomorphic to $SL_3$ or to $Sp_2$.

In detail, such a $G$ contains a subgroup $G_0$ which is $split$ over $\mathbb Q$ and of the same $\mathbb Q$-rank as $G$. By looking at the Dynkin diagram of $G_0$, one can extract a sub-diagram of type $A_2$ or $B_2$ except for $G_2$ where this (i.e. that $G_2$ contains $A_2$, the root system of short roots in the root system of $G_2$) can be proved directly by looking at its root system.

[Edit] Misha (in the comments) was right; this result that any $k$-simple group of $k$-rank at least two contains a subgroup locally isomorphic to $SL_3$ or $Sp_4$) is explicitly stated and proved in Margulis' book; see Proposition (1.6.2) of Margulis' book titled " Discrete subgroups of Semi-simple Lie groups (Ergbnisse tract, volume 17)". The result about arithmetic groups can be deduced from this by taking $k={\mathbb Q}$.

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  • $\begingroup$ Note that you only address the "if" implication here. $\endgroup$
    – YCor
    Jan 14, 2014 at 15:26
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    $\begingroup$ you are right; but you have already addressed the "only if" part in the comments above. $\endgroup$ Jan 14, 2014 at 15:29
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    $\begingroup$ Note that the Bourbaki talk (but probably not Kazhdan's paper) is available online: numdam.org/numdam-bin/fitem?id=SB_1966-1968__10__507_0 $\endgroup$ Jan 14, 2014 at 23:31
  • $\begingroup$ @Jim Thank you very much for the link to that paper; I had an old xerox copy of the Bourbaki talk. $\endgroup$ Jan 15, 2014 at 1:04
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    $\begingroup$ In Kazhdan's paper, however, he does not prove the result for which the OP has asked; he seems to prove property T only for simple groups of real rank at least three. It is only in Kirillov-Delaroche paper that I find property (T) proved for all simple lie groups of real rank at least two $\endgroup$ Oct 13, 2014 at 0:44

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