Multi-orderable groups

A Cayley object for a group G is a structure M on G such that the right regular action of G gives an automorphism group of M. (Thus a Cayley graph is a Cayley object which happens to be a graph.) A relational structure is homogeneous if every isomorphism between finite substructures can be extended to an automorphism of M. As part of a general investigation of homogeneous Cayley objects for countable groups, I conjectured in 2000 that the universal homogeneous n-tuple of total orders is a Cayley object for the free abelian group of rank m if and only if m > n. I have just succeeded in proving this conjecture, using a theorem of Kronecker on diophantine approximation. The talk will be self-contained and all concepts will be explained.