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

TL;DR

This paper studies the growth rates of group actions on products of Hadamard spaces, showing continuity properties, addressing open questions, and extending invariant densities under certain conditions.

Contribution

It establishes the continuity of the growth rate function and provides a negative answer to an open question, extending the theory of invariant densities for group actions.

Findings

Growth rate $ heta$ is continuous in the slope under mild conditions.

Negative answer to G. Link's open question in general.

Extension of $(b, heta)$-densities under certain conditions.

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.