The Brauer trees of non-crystallographic groups of Lie type

In this article we determine the Brauer trees of the unipotent blocks with cyclic defect group in the `groups' $I_2(n,q)$, $H_3(q)$ and $H_4(q)$. The degrees of the unipotent characters of these objects were given by Lusztig, and using the general theory of perverse equivalences we can reconstruct the Brauer trees that would be consistent with Deligne--Lusztig theory and the geometric version of Brou\'e's conjecture. We construct the trees using standard arguments whenever possible, and check that the Brauer trees predicted by Brou\'e's conjecture are consistent with both the mathematics and philosophy of blocks with cyclic defect groups.