Group rings and hyperbolic geometry

Abstract

For a group acting on a hyperbolic space, we set up an algorithm in the group algebra showing that ideals generated by few elements are free, where few is a function of the minimal displacement of the action, and derive algebraic, geometric, and topological consequences. In particular, we obtain lower bounds on Morse complexity of closed hyperbolic manifolds in terms of injectivity radius.

29 pages, improved injectivity radius hypothesis from $n^2$ to $\log (n!)$, to appear in Geometry and Topology