Spaces and groups with conformal dimension greater than one.

The conformal dimension of a metric space is a quasi-symmetric invariant that in some sense measures the `best shape’ of the metric space under quasi-symmetric deformations. In this talk I’ll survey some known results about conformal dimension and give examples where this invariant is interesting, such as the boundary at infinity of a Gromov hyperbolic group, paying particular attention to spaces of topological dimension one.

I’ll also describe recent work that gives a lower bound greater than one for a natural class of metric spaces that includes boundaries of hyperbolic groups that are connected with no local cut points.