Some infinite groups generated by involutions have Kazhdan's property (T) (Q2753163)

Using a nice construction which develops an idea of \textit{M. Bourdon} [Ergodic Theory Dyn. Syst. 20, No. 2, 343-364 (2000; Zbl 0965.53030)], the author produces a large family of infinite groups generated by involutions. The construction starts with a finite group action on a bipartite graph with certain properties. The cone over this graph is then regarded as a rigid \(2\)-orbihedron. The properties of the action on the graph force this orbihedron to be nonpositively curved, so that its universal cover \(X\) is contractible. Half of the vertices of the graph produce reflections on \(X\), and the other half lift to points fixed by several of these reflections. For many of the resulting reflection groups, the lowest eigenvalue of the graph-theoretic Laplacian of the link of each vertex of \(X\) is greater than \(1/2\), showing that unlike Coxeter groups, these groups satisfy Kazhdan's property (T).