Splitting in map groups

Abstract

Every locally compact maximally almost periodic group G has a normal vector subgroup, the centralizer of which is of finite index. This vector subgroup is nontrivial whenever the identity component of G is not compact. Furthermore, if G has relatively compact conjugacy classes, then G ≅ R n × L G \cong {R^n} \times L where L has a compact open normal subgroup. Several structure theorems are also obtained for cases in which splitting need not occur.