Simple dynamics of automorphisms (Q5928494)

A negatively curved (word hyperbolic) group \(G\) acts on its Gromov boundary \(\partial G\) as a (discrete) convergence group (also, the left action of \(G\) coincides with that of its inner automorphism group). A convergence group is a group of homeomorphisms having the main dynamical property of groups of Möbius transformations, allowing the distinction of its elements into elliptic, parabolic and loxodromic ones. Equivalently, a group acts as a convergence group on a compact first countable space \(X\) if and only if it acts properly discontinuously on the subspace \(\mathcal T\) of \(X^3\) consisting of all triples of points which are pairwise different.NEWLINENEWLINENEWLINEThe question considered in the present paper is the following: which other subgroups of the automorphism group of \(G\) have a convergence action on \(\partial G\)? The basic case of free groups is considered. The main result states that, if a group of automorphisms of a free group \(F\) contains the inner automorphisms group as a normal subgroup and acts as a convergence group on \(\partial F\) then it is a finite extension of the inner automorphism group. For the proof the result is used that if a convergence group of the first kind (i.e. the ordinary set is empty) has a normal subgroup acting cocompactly on \(\mathcal T\) then the normal subgroup has finite index. Various concrete examples of automorphisms of a free group are discussed which do not generate convergence groups on its boundary.