A fixed point theorem of discrete group actions on Riemannian manifolds
A group \(\Gamma\) has property \((F)\) iff any isometric action of \(\Gamma\) on a manifold \(N\) of nonpositive sectional curvature has a fixed point in the compactification \(\overline N\). The author proves the following theorem: If \(\Sigma\) is a finite simplicial complex with an admissible weight \(c\) such that \(\lambda_{1,\text{loc}}(\Sigma,c)>{1\over 2}\), then the fundamental group of~\(\Sigma\) has property \((F)\). Here, an admissible weight is a metric on~\(C^*(\Sigma)\) of a certain type, and the condition~\(\lambda_{1,\text{loc}}(\Sigma,c)>{1\over 2}\) refers to the Laplacian on \(0\)-forms on the links of vertices in \(\Sigma\). The proof uses harmonic maps from simplical complexes to manifolds. The theorem above implies the superrigidity theorem of \textit{G. A. Margulis} [`Discrete subgroups of semisimple Lie groups', Springer (1991; Zbl 0732.22008)] for cocompact lattices in \(p\)-adic groups of rank greater than one, at least if the residue field is sufficiently large.
scientific article; zbMATH DE number 1368005