On Sidki's presentation for orthogonal groups

Abstract

We study presentations, defined by Sidki, resulting in groups $y(m,n)$ that are conjectured to be finite orthogonal groups of dimension $m+1$ in characteristic two. This conjecture, if true, shows an interesting pattern, possibly connected with Bott periodicity. It would also give new presentations for a large family of finite orthogonal groups in characteristic two, with no generator having the same order as the cyclic group of the field. We generalise the presentation to an infinite version $y(m)$ and explicitly relate this to previous work done by Sidki. The original groups $y(m,n)$ can be found as quotients over congruence subgroups of $y(m)$. We give two representations of our group $y(m)$. One into an orthogonal group of dimension $m+1$ and the other, using Clifford algebras, into the corresponding pin group, both defined over a ring in characteristic two. Hence, this gives two different actions of the group. Sidki's homomorphism into $SL_{2^{m-2}}(R)$ is recovered and extended as an action on a submodule of the Clifford algebra.