  The notion of almost centralizer and almost commutator are introduced and
basic properties are established. They are used to study $\widetilde{\mathfrak
M}\_c$-groups, i. e.groups for which every descending chain of centralizers
each having infinite index in its predecessor stabilizes after finitely many
steps. The Fitting subgroup of such groups is shown to be nilpotent and a
theorem of Hall for nilpotent groups is generalized to ind-definable almost
nilpotent subgroups of $\widetilde{\mathfrak M}\_c$-groups.
