Remarks on the actions of some groups on the product of Hadamard spaces

Abstract

For a product of Hadamard spaces $X=X_1\times X_2$ on which some group $G\subset Is(X_1)\times Is(X_2)$ acting, G. Link arXiv:1107.3755v1 introduced the growth rate $\delta_{\theta}$ of slope $\theta$ to construct a $G-$invariant $(b,\theta)$-density. First, we show that $\delta_{\theta}$ is continuous in the slope $\theta$ as the above group action with some mild condition. Second, we give a negative answer to a question raised by G. Link in \cite{1} in general. And further results about the question are discussed. In the end, we can extent the $(b,\theta)$-densities in some reasonable condition.