Coarse fixed point theorem (Q1046449)

The article under review is an announcement of a couple of results about the action of abelian groups on coarse metric spaces. So, given is a proper metric space \(X\), a finitely generated abelian group \(G\) (with word metric) acting on \(X\) by coarse maps. One then has an induced action by homeomorphisms on the Higson corona of \(X\). The main result states that if there is one orbit \(Gx\) such that the map \(g\mapsto gx\) is coarse then the induced action on the Higson corona has a fixed point. As a corollary, a coarse version of Browder's fixed point theorem is derived (for the union of \(X\) and its Higson corona). It is stated that proofs are to be given elsewhere.