For orientable connected closed manifolds of the same
dimension, there is a kind of order: M DOMINATES N if there exists a
continuous map of non-zero degree from M onto N. In a first part, I will
recall the notion of degree (Brouwer, Hopf), show some examples of
(non-)domination, and in particular discuss when a manifold can (or
cannot) be dominated by a product. These considerations suggest a notion
for groups (fundamental groups): a group is PRESENTABLE BY A PRODUCT if
it contains two infinite commuting subgroups which generate a subgroup
of finite index. In a second part, I plan to discuss groups
(non-)presentable by products, including some Coxeter groups.