## Probability Seminar: Juan Souto

- Date: 03/21/2012
- Time: 15:00

University of British Columbia

Distributional limits of Riemannian manifolds and graphs with sublinear genus growth

Abstract

Studying Gromov-Hausdorff limits of sequences of Riemannian manifolds

$(M_i)$ satisfying suitable conditions on their local geometry is an

extremely fruitful idea. However, in the most interesting case that the

diameter of $M_i$ grows without bounds, one is forced to choose base

points $p_i\in M_i$ and consider limits of the pointed spaces

$(M_i,p_i)$ in the pointed Gromov-Hausdorff topology. The choice of the

base points $p_i$ influences enormously the obtained limits. Benjamini

and Schramm introduced the notion of distributional limit of a sequence

of graphs; this basically amounts to "choosing the base point by

random". In this talk I will describe the distributional limits of

sequences $(M_i)$ of manifolds with uniformly pinched curvature and

satisfying a certain condition of quasi-conformal nature. I will also

explain how these results yield a modest extension of Benjamini's and

Schramm's original result. This is joint work with Hossein Namazi and

Pekka Pankka.