Hyperbolic elements in negatively curved groups
The paper shows that, in a negatively curved group \(G\) (= Gromov hyperbolic group, see \textit{M. Gromov} [Publ., Math. Sci. Res. Inst. 8, 75-263 (1987; Zbl 0634.20015)]), the conjugacy class of any infinite cyclic subgroup contains a straight element, that is an element \(g\in G\) with \(|g^n|=n|g|\). To prove this fact, the author uses \textit{J. Cannon's} approach [in Hyperbolic geometry and ergodic theory 1989, 315-369 (1991; Zbl 0764.57002)] to negatively curved groups and Cannon's results on cone types and quasigeodesics in such groups. As a corollary, it is obtained that the translation number of an element of a negatively curved group is rational with uniformly bounded (in the group) denominator. This implies that the endpoints of conjugates of a hyperbolic element are dense in the boundary of the group. This resembles the situation with the limit set of Kleinian groups and allows the author to get an extension of another result from Kleinian groups, namely that every normal subgroup of a negatively curved group is either elementary (finite) or has the same boundary at infinity as the group \(G\) itself. The author also proves the double density theorem of Gromov (see the above reference) that any two boundary points in \(\partial G\) are ``almost'' the endpoints of a hyperbolic element.
scientific article; zbMATH DE number 806229