Abstract

Let P be a numerical invariant of a topological space, let X be a space, and let (X_i) be a (residual) tower of covers of finite degree d_i. The P-gradient associated to the pair X, (X_i) is the limit of P(X_i) / d_i as i tends to infinity. The three most important questions to answer when studying P-gradients are: Does the limit exist? Does it depend on the choice of tower? Can we relate the limit to another known invariant? When P is the nth rational Betti number, the answer to all these questions is provided by Lück’s celebrated Approximation Theorem, which states that the limit always equals the nth L^2 Betti number. In this talk, we will discuss several gradient invariants and the many open problems in this area. We will focus primarily on the mod-p homology gradients, where our main result will be a computation of these invariants for Bestvina—Brady groups, and more generally kernels of maps from Raags to Z. If time permits, we will also mention a connection with algebraic fibring. This is joint work with Sam Hughes and Ian Leary.