Some properties of isometry groups of pinched manifolds (Q2767346)

This paper studies the isometry group \(M\left( X\right) \) of a pinched manifold \(M\), which generalizes the important Möbius group (for \(M=H^{n}\)). The author derives Jorgensen type of inequalities, like \(\max \left\{ n_{f}\left( x\right) ,n_{\left[ f,g\right] }\left( x\right) \right\} \geq 0.49\) for \(x\in X\) when \(<f,g>\) is a discrete non-elementary subgroup with \( f\in M\left( X\right) \) non-elliptic, and shows that a non-elliptic subgroup \(G\) of \(M\left( X\right) \) is non-elementary if and only if elements of \( \left[ G,G\right] \) have no common fixed points. Furthermore some criteria for a subgroup \(G\) of \(M\left( X\right) \) to be discrete are presented.