The authors claim that any group acting non-trivially on finitely many Gromov-hyperbolic spaces contains simultaneously hyperbolic elements with positive density, but a central lemma in the proof fails for non-orientable lineal actions.
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A general construction of simultaneously hyperbolic elements
The authors claim that any group acting non-trivially on finitely many Gromov-hyperbolic spaces contains simultaneously hyperbolic elements with positive density, but a central lemma in the proof fails for non-orientable lineal actions.