# Characterisation of Q-rank 1

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 $SO(2,3)_{\Bbb Z}$.

• 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 $SL_3$. – YCor Oct 7 '13 at 13:35
• 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. – YCor Oct 7 '13 at 13:51
• This appears in Witte-Morris' book about arithmetic-groups - people.uleth.ca/~dave.morris/books/IntroArithGroups.pdf, see ch9 section H in there. – Asaf Oct 7 '13 at 15:06
• Unfortunately that chapter 9 is unfinished and has no references yet. – ThiKu Oct 8 '13 at 5:18
• I think it is in Margulis' book. – Misha Oct 8 '13 at 12:05

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.

 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 $k={\mathbb Q}$.

• Note that you only address the "if" implication here. – YCor Jan 14 '14 at 15:26
• you are right; but you have already addressed the "only if" part in the comments above. – Venkataramana Jan 14 '14 at 15:29
• 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 – Jim Humphreys Jan 14 '14 at 23:31
• @Jim Thank you very much for the link to that paper; I had an old xerox copy of the Bourbaki talk. – Venkataramana Jan 15 '14 at 1:04
• 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 – Venkataramana Oct 13 '14 at 0:44