Soren Galatius (Stanford)
Simplicial deformation rings and the homology of arithmetic groups

Abstract: A classical result of Mazur asserts that any absolutely irreducible representation \rho of a group (or pro-group) over a field admits a universal deformation. This is a representation over a complete local ring, whose reduction modulo the maximal ideal is identified with \rho, and which is in a suitable sense universal with these properties. I will discuss joint work with Akshay Venkatesh, in which we enrich this picture to simplicial rings. The classical deformation ring is recovered by taking \pi_0, but some higher homotopy groups may be non-trivial. Subject to some (partially established) conjectures, we prove that homology of certain arithmetic groups admit the structure of a free module structure over the homotopy groups of the simplicial deformation ring for certain Galois representations.