#### Vol. 294, No. 1, 2018

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Simultaneous construction of hyperbolic isometries

### Matt Clay and Caglar Uyanik

Vol. 294 (2018), No. 1, 71–88
##### Abstract

Given isometric actions by a group $G$ on finitely many $\delta$-hyperbolic metric spaces, we provide a sufficient condition that guarantees the existence of a single element in $G$ that is hyperbolic for each action. As an application we prove a conjecture of Handel and Mosher regarding relatively fully irreducible subgroups and elements in the outer automorphism group of a free group.

##### Keywords
hyperbolic isometries, free groups, fully irreducible
Primary: 20F65