Separation profiles of graphs of fractals

TL;DR

This paper investigates the separation profiles of spheres in hyperbolic graphs with fractal boundaries, revealing that the separation scales with a dimension smaller than the conformal dimension, contrasting with symmetric spaces.

Contribution

It computes separation profiles for fractal boundaries like Sierpiński carpets and Menger sponges, linking them to known bounds on conformal dimension.

Findings

Separation scales as n^{(d-1)/d} for some d<conformal dimension

d matches known lower bounds on conformal dimension

Contrasts with behavior in rank 1 symmetric spaces

Abstract

We continue the exploration of the relationship between conformal dimension and the separation profile by computing the separation of families of spheres in hyperbolic graphs whose boundaries are standard Sierpi\'nski carpets and Menger sponges. In all cases, we show that the separation of these spheres is $n^{\frac{d-1}{d}}$ for some $d$ which is strictly smaller than the conformal dimension, in contrast to the case of rank 1 symmetric spaces of dimension $\geq 3$. The value of $d$ obtained naturally corresponds to a previously known lower bound on the conformal dimension of the associated fractal.