Variations on a Theme of Borel : An Essay on the Role of the Fundamental Group in Rigidity

In the middle of the last century, after hearing a talk of Mostow on one of his rigidity theorems, Borel conjectured in a letter to Serre a purely topological version of rigidity for aspherical manifolds (i.e. manifolds with contractible universal covers). The Borel conjecture is now one of the central problems of topology with many implications for manifolds that need not be aspherical. Since then, the theory of rigidity has vastly expanded in both precision and scope. This book rethinks the implications of accepting his heuristic as a source of ideas. Doing so leads to many variants of the original conjecture - some true, some false, and some that remain conjectural. The author explores this collection of ideas, following them where they lead whether into rigidity theory in its differential geometric and representation theoretic forms, or geometric group theory, metric geometry, global analysis, algebraic geometry, K-theory, or controlled topology. "This essay is a work of historical fiction - the "What if Eleanor Roosevelt could fly?" kind. The Borel conjecture is a central problem in topology: it asserts the topological rigidity of aspherical manifolds (definitions below!). Borel made his conjecture in a letter to Serre some 65 years ago, after learning of some work of Mostow on the rigidity of solvmanifolds. We shall re-imagine Borel's conjecture as being made after Mostow had proved the more famous rigidity theorem that bears his name - the rigidity of hyperbolic manifolds of dimension at least three - as the geometric rigidity of hyperbolic manifolds is stronger than what is true of solvmanifolds, and the geometric picture is clearer"-- Provided by publisher Shmuel Weinberger describes here analogies between geometric topology, differential geometry, group theory, global analysis, and noncommutative geometry. He develops deep tools in a setting where they have immediate application. The connections between these fields enrich each and shed light on one another.