Ergodic Theory and Geometry of Nilpotent Groups

We investigate the ergodic theory and geometry of finitely generated virtually nilpotent groups. Let Γ be a finitely generated virtually nilpotent group. In the first part of the thesis, we consider three closely related problems: (i) convergence to a deterministic asymptotic cone for an equivariant ergodic family of inner metrics on Γ, generalizing a theorem of Pansu; (ii) the asymptotic shape theorem for First Passage Percolation for general (not necessarily independent) ergodic processes on edges of a Cayley graph of Γ; (iii) the sub-additive ergodic theorem over a general ergodic Γ-action. The limiting objects are given in terms of a Carnot-Caratheodory metric on the graded nilpotent group associated to the Mal’cev completion of Γ. In the second part of the thesis we prove an analog for integrable measurable cocycles of Pansu’s differentiation theorem for Lipschitz maps between Carnot-Caratheodory spaces. This yields an alternative, ergodic theoretic proof of Pansu’s quasi-isometric rigidity theorem for nilpotent groups, answers a question of Tim Austin regarding integrable measure equivalence between nilpotent groups, and gives an independent proof and strengthening of Austin’s result that integrable measure equiv- alent nilpotent groups have bi-Lipschitz asymptotic cones. The main tools for this part are a nilpotent-valued cocycle ergodic theorem and a Poincare recurrence lemma for nilpotent groups, which may be of independent interest.