Reachability problems in quaternion matrix and rotation semigroups

We examine computational problems on quaternion matrix and ro- tation semigroups. It is shown that in the ultimate case of quaternion matrices, in which multiplication is still associative, most of the decision problems for ma- trix semigroups are undecidable in dimension two. The geometric interpretation of matrix problems over quaternions is presented in terms of rotation problems for the 2 and 3-sphere. In particular, we show that the reachability of the ro- tation problem is undecidable on the 3-sphere and other rotation problems can be formulated as matrix problems over complex and hypercomplex numbers.