Inertial properties in groups

Let G be a group and p be an endomorphism of Gý. ýA subgroup H of G is called p-inert if Hp∩H has finite index in the image Hpý. ýThe subgroups that are p-inert for all inner automorphisms of G are widely known and studied in the literatureý, ýunder the name inert subgroupsý.
ýThe related notion of inertial endomorphismý, ýnamely an endomorphism p such that all subgroups of G are p-inertý, ýwas introduced in \cite{DR1} and thoroughly studied in \cite{DR2,DR4}ý. ýThe ``dualý" ýnotion of fully inert subgroupý, ýnamely a subgroup that is p-inert for all endomorphisms of an abelian group Aý, ýwas introduced in \cite{DGSV} and further studied in \cite{Chý, ýDSZ,GSZ}ý. ýThe goal of this paper is to give an overview of up-to-date known resultsý, ýas well as some new onesý, ýand show how some applications of the concept of inert subgroup fit in the same picture even if they arise in different areas of algebraý. ýWe survey on classical and recent results on groups whose inner automorphisms are inertialý. ýMoreoverý, ýwe show howý ýinert subgroups naturally appear in the realm of locally compact topological groups or locally linearly compact topological vector spacesý, ýand can be helpful for the computation of the algebraic entropy of continuous endomorphismsý.