I'm interested in the subgroup of the indefinite orthogonal group consisting of integer matrices i.e. $$ZO(p,q):=\{M \in GL(p+q, \mathbb{Z})| M^TI_{p,q}M=I_{p,q}\},$$ where $I_{p,q}$ denotes the diagonal matrix with p 1's and q -1's on the diagonal.
It is not hard to show that $ZO(p,0)$ and $ZO(0,q)$ are finite. My question is: When $p,q>0$ are these groups infinite? I'm particularly interested in the case $q=1$.
My suspicion is that $ZO(p,q)$ is infinite in most cases. However I'm unable to prove it. When $p,q>0$, One can find plenty of integer vectors which satisfy $v I_{p,q} v^T=\pm 1$, however my difficulty has been finding orthogonal vectors with this property.
As I'm sure these groups must have been studied before, any direction on where to look in the literature would also be appreciated.
Thanks.
If $q=1$ and $p>1$ the group is infinite, simply because it then contains a matrix with an $n-3$ square identity block and a 3 by 3 block with all positive entries, matrix B from http://en.wikipedia.org/wiki/Tree_of_Pythagorean_triples.
Your objects are called odd unimodular Lorentzian lattices. The notation in SPLAG is $I_{n,1}$ with $n=p.$ SPLAG is Sphere Packing, Lattices, and Groups, by Conway and Sloane. You want chapters 26 and 27 especially. Here we go, Coxeter and Whitrow found the automorphism group of $I_{3,1}$ as a hyperbolic reflection group. Vinberg showed that the subgroup generated by reflections has finite index only if $n \leq 19,$ and confirmed that it does so if $n=9,17.$ The algorithm for dealing with these things is called Vinberg's algorithm.
So, some of the phrases that go with this are "Coxeter groups" and "reflection groups."
This looks promising, a book by Coxeter and Moser, Generators and Relations for Discrete Groups.