Inertial properties in groups

‎‎Let G be a group and φ be an endomorphism of G. A subgroup H of G is calledφ-inert if the meet of Hf and H has finite index in the image H^φ. The subgroups that are φ-inert for all inner automorphisms of G are widely known and studied in the literature, under the name inert subgroups. The related notion of inertial endomorphism, namely an endomorphism f such that all subgroups of G are φ-inert, was introduced in [34] and thoroughly studied in [37, 35]. The dual notion of fully inert subgroup, namely a subgroup that is φ-inert for all endomorphisms of an abelian group A, was introduced in [51] and further studied in [25, 54, 76]. The goal of this paper is to give an overview of up-to-date known results, as well as some new ones, and show how some applications of the concept of inert subgroup fit in the same picture even if they arise in different areas of algebra. We survey on classical and recent results on groups whose inner automorphism are inertial. Moreover, we show how inert subgroups naturally appear in the realm of locally compact topological groups or locally linearly compact topological vector spaces, and can be helpful for the computation of the algebraic entropy of continuous endomorphisms.