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Separation profiles of graphs of fractals. (English) Zbl 07487129

Summary: 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 Sierpiński 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 \(\geq3\). The value of \(d\) obtained naturally corresponds to a previously known lower bound on the conformal dimension of the associated fractal.

MSC:

20F65 Geometric group theory
20F67 Hyperbolic groups and nonpositively curved groups
05C40 Connectivity
05C63 Infinite graphs
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