Orthogonal group of quadratic form

Question

Orthogonal group of the quadratic form over fields, somehow, is well-studied. Indeed E. Cartan has proved for quadratic forms over the reals or complexes that any orthogonal transformation is a product of at most $n$ symmetries, where $n$ is the dimensionality of the underlying vector space. This result was generalized by Dieudonne to quadratic forms over arbitrary base fields.

One can try to understand the integral orthogonal group of an integral quadratic form, more precisely:

Question: Let $q(x_1,\dots,x_n)$ be an integral quadratic form, can we say the integral orthogonal group, denoted as usual by $O_\mathbb{Z}(q)$, is finitely generated. If this would be the case, what can we say about the number of generator?

Let me pick the following special example: $$ q(x,y,z)=x^2+y^2-z^2 $$ One can show $O_\mathbb{Z}(q)$ acts transitively on $$ \{(x,y,z)\in \mathbb{Z}^3: q(x,y,z)=0\} $$ So understanding the number of generators of $O_\mathbb{Z}(q)$, could gives us the space of integral solutions of $q(x,y,z)=0$. For instance, Keith Conrad has a wonderful note entitled with "Orthogonal group of $x^2 + y^2 - z^2$", who proved $O_\mathbb{Z}(q)$ is generated by five elements (I think it was known before but Conrad exposition is great).

This example shows, that the above question can be interesting. What do we know bout the about question?

Answer 1

Yes, these groups are arithmetic lattices, and are therefore finitely generated. I believe Selberg showed that they are cofinite volume (with respect to the discrete action on the appropriate symmetric space). When the form is definite, it is a finite group. When the form is Lorentzian, the group may be shown to have a nice Ford domain with respect to its action on hyperbolic space, which shows that it is finitely generated. In principle, using the volume computation, one could give an upper bound on the number of generators in this case. For small examples, I think that the number of generators grows linearly with the dimension. Also, these groups are not generated by reflections for high enough dimensions by a result of Nikulin (even up to finite-index).

When the form has rank $>1$, then the group has property (T), and therefore is finitely generated by a result of Kazhdan. The original proof though appears to be due to Borel-Harish Chandra. I think this may also be proved using the Borel-Serre compactification. For information about arithmetic groups, check out the book (in progress) by Dave Witte-Morris. In this case, the groups are generated by reflections up to finite index. Venkataramana has some results on the number of generators of such lattices and finite-index subgroups.