Finite subgroups of Spin(9)

I'm trying to classify compact manifolds $M^{16}$ with a metric which is locally conformal to a (local) metric with holonomy (included in) Spin(9)$\subset$SO(16). To do this, I would need a complete list of finite subgroups of Spin(9) acting freely on $S^{15}$. Any hint on finite subgroups of Spin(9) (even not acting freely) would be very helpful!

2 Answers

Although there exists an algorithm which will list all finite subgroups of $Spin(9)$ I suspect it is not effective. For background and references see

The finite subgroups of SU(n)

I don't know if imposing the condition that the group acts freely on $S^{15}$ improves the situation.

• the fact that the action is free should surely simplify matters. after all there is a classification of spherical spaceforms by Wolf and one can presumably check which of the groups on his list are conjugate to subgroups of $Spin(9)$. This hardly sounds pleasant though. Jul 13, 2012 at 19:27
• Thank you for pointing me to the cited thread. That thread is very informative and interesting. Unfortunately, the Zassenhaus paper they refer to is in German, making it inaccessible to me. Does anybody know if that Zassenhaus paper is translated or otherwise explained in english somewhere? I also checked the Wolf's book, but it is really not an easy task to check the conjugacy class using his list. Aug 3, 2012 at 14:50
• I don't know of another account. This is not regarded as an effective algorithm. Aug 4, 2012 at 16:59

After some thinking, I came up with the following very partial answer, that I put here as a reference.

Consider the diagonal inclusion of $\text{Sp}(1)$ in $\text{Spin}(9)$ given by sending a unitary quaternion $q\in\text{Sp}(1)$ into the map $L_q\in\text{Spin}(9)\subset\text{SO}(16)$, where $L_q$ is defined by $$L_q:\mathbb{H}^4\longrightarrow \mathbb{H}^4,\qquad L_q(h_1,h_2,h_3,h_4)=(qh_1,qh_2,qh_3,qh_4).$$

If $q\neq 1$, any such $L_q$ acts without fixed points on $S^{15}$. Since the finite subgroups $G$ of $\text{Sp}(1)$ are known (polyhedrons classification), any such $G$ gives a finite subgroup $L_q(G)$ of $\text{Spin}(9)$ acting without fixed points.