Spaces and groups with conformal dimension greater than one

We show that if a complete, doubling metric space is annularly linearly connected then its conformal dimension is greater than one, quantitatively. As a consequence, we answer a question of Bonk and Kleiner: if the boundary of a one-ended hyperbolic group has no local cut points, then its conformal dimension is greater than one.

💡 Research Summary

The paper establishes a robust lower bound for the conformal dimension of a broad class of metric spaces and applies this result to the boundaries of hyperbolic groups. The main theorem states that any complete metric space that is both doubling (i.e., every ball can be covered by a uniformly bounded number of balls of half the radius) and annularly linearly connected (ALC) has conformal dimension strictly greater than one. Moreover, the authors provide a quantitative estimate: the excess over one depends only on the doubling constant and the ALC constant, not on any finer structure of the space.

The proof proceeds by exploiting the relationship between conformal dimension and the p‑modulus of curve families, a technique pioneered by Tyson. In an ALC space one can construct, for each annulus, a continuum joining any two points in the inner component whose diameter is controlled by a fixed multiple λ of the distance between the points. By chaining such continua across a nested sequence of annuli, the authors build a global family of curves that “fills” the space at all scales. Careful combinatorial estimates show that for every p in an interval