A general construction of simultaneously hyperbolic elements

Abstract: In this paper, we give an explicit construction of simultaneously hyperbolic elements in a group acting on finitely many Gromov-hyperbolic spaces under the weakest conditions. This essentially generalizes results of Clay-Uyanik in \cite{CU18}, of Genevois in \cite{Gen19}, and of Balasubramanya-Fern\'{o}s in \cite{BF24}. Besides, we show that the set of simultaneously hyperbolic elements has strictly positive density with respect to any proper word metric under the weakest conditions. This recovers many classical counting results, eg. the main result of Wiest in \cite{Wie17}. As an important ingredient in the proof of main results, we show that the set of simultaneously contracting elements in a group acting on finitely many metric spaces with contracting property has strictly positive density with respect to any proper word metric. This generalizes two results of Wan-Xu-Yang in \cite{WXY24} and of Balasubramanya-Fern\'{o}s in \cite{BF24}.